Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M.Thoma
8
Ruth F. Curtain Anthony J. Pritchard
Infinite Dimensional
Linear Systems Theory
Springer-Verlag Berlin Heidelberg New York 1978
Series Editors A. V. Balakrishnan • M. Thoma Advisory Board A. G..1. MacFarlane • H. Kwakernaak • Ya. Z. Tsypkin Authors Dr. Ruth F. Curtain Control Theory Centre, University of Warwick presently at: Rijksuniversiteit te Groningen Mathematisch Instituut Groningen Netherlands Dr. Anthony .I. Pritchard Director, Control Theory Centre University of Warwick Coventry CV4 ?AL United Kingdom
ISBN 3-540-08961-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08961-6 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Reltz Offsetdruck, Hemsbach/Rergstr. 206113020-543210
PREFACE
Over the past five years the major research effort of the Control Theory Centre at the University of Warwick has been in the area of infinite dimensional system theory.
The p h i l o s o p h y underlying the re-
search has been to develop a mathematical
framework which enables the
generalisation of the finite dimensional results to infinite dimensions and which includes both distributed parameter systems and differential delay systems as special cases.
So following the lead of Fattorinl
in
[6], we describe the system dynamics in terms of a strongly continuous semigroup on an appropriate Banach space.
Using this unifying mathem-
atical approach it is possible to clarify the essential concepts of observability,
controllability,
the quadratic cost control problem,
the estimation and control problems
for stochastic systems.
and
At this
stage we feel it is appropriate to present the culmination of this collective effort in a coordinated way in the form of lecture notes. Of course it has not been possible to cover all of the systems theory concepts,
and significant omissions are realization theory and identif-
ication.
Readers interested in these areas should see for instance
and watch out for the forthcoming monograph by Fuhrmann
[7].
[ll]
We hope,
however, that these notes will provide an introduction to infinite dimensional system theory,
accessible to readers with a knowledge of fin-
ite dimensional theory and some functional
analysis.
The treatment of
the material reflects our own personal approach and is by no means the only way or even the most commonly accepted way.
While it is aesthet-
ically pleasing that the abstract formulation yields results which mirror those for finite dimensions;
because the state space is infinite
dimensional this superficial resemblance can be misleading.
Consequen-
tly we have examined the implications of the abstract theory to specific examples of distributed and delay systems. Many other researchers have contributed to infinite dimensional systems theory, some using a semigroup approach and others using methods appropriate for special classes of systems. and Lions
For example Butkovskii
[9] have examined optimal control problems
distributed parameter systems and Bensoussan of stochastic distributed systems.
[2]
for deterministic
[I] estimation
and control
The pioneering work on controllab-
ility and observability of hyperbolic and parabolic partial differential equations was developed by Fattorini ova
[8], Manitius
[iO], Banks
[4]
[5] and Russell[12],whilst
and Lee
[3]
Kirill-
have contributed to the
study of differential delay systems.
It has not been possible to incl-
ude all of these special results for particular systems in these notes. What we have tried to do, however,
is to illustrate through the examples
how many of these special results may be obtained using an abstract semigroup approach and at the end of each chapter included references of other contributions. The first chapter reviews the types of finite dimensional systems theory results which will be generalized to infinite dimensions in chapters 3-7 and gives several examples of how such problems may arise in distributed and delay systems.
To make these notes self-contained,
chapter 2 presents the known results on semigroups, which we shall need in later chapters.
Chapters 3-7 form the core of the book, namely the
extension of the finite dimensional systems theory results outlined in Chapter 1 for time-invariant linear systems.
All these results are
proved in detail and are illustrated by several examples and would be appropriate for an introductory graduate course in linear systems theory. Chapters 8 & 9 are concerned with extensions of the results of Chapters 3-7 to more complicated systems, namely time dependent systems and distributed systems allowing for boundary control and point observations. The difficulties which arise in trying to extend the results to more general systems are technical mathematical ones rather than conceptual ones.
As the technical details are already available in the literature,
we have chosen to motivate the approach using simple examples and have omitted proofs which are heavy technical extensions of those in earlier chapters.
So although a complete presentation of the results is avail-
able in Chapters 8 and 9, many proofs are given in outline only.
Again,
considerable attention is given t o analyzing the implications for distributed and delay systems by means of examples. These lecture notes have been influenced by the many visitors to the Control Theory Centre and especially by the SRC
funded research fellows,
S.P. Banks, A. Ichikawa, E.P. Ryan, R. Triggiani, A. Wirth and J. Zabczyk. It is also a pleasure to pay tribute to the former directors of the Centre, Professor L. Markus and Professor P.C. Parks, for their guidance
Science Research Council
(United Kingdom) which has supported the
Control Theory Centre from its inception in 1970.
V and encouragement Professor J.A.
in our research,
Shercliff
tinuing support of the Centre. for transforming
and to Professor J.L. Douce and
of the Engineering
Department
Finally special
a largely illegible manuscript
for their con-
thanks to Frances Ryan into a respectable
typescript. A.J. Pritchard,
Director of Control Theory Centre, University of Warwick.
Ruth F. Curtain,
Control Theory Centre, University of Warwick. (presently at: Mathematics Institute, Rijksuniversiteit Groningen.)
References [i]
Bensoussan,
A.
Filtrage
optimal des systemes
lineaires,
Dunod
1971. [2]
Butkovskii,
[3]
Chyung,
[4]
Eller,
A.G.
Theory of optimal meter systems,
D.H. and Lee, E.B.
American Elsevier,
para-
1969.
Delayed action control problems,
Automatica, D.H., Aggarwal,
control of distributed
6 (1970), pp.395-4OO.
J.K.
and Banks,
linear time-delay
H.T.
systems,
Optimal control of IEEE Trans., AC 14
(1969), pp.678-687. [5]
Fattorini,
H.O. On complete
controllability
J. Diff. Eqns., [6]
Fattorini,
H.O. Time optimal
of linear systems,
3 (1967), pp.391-402.
control of solutions
differential
equations,
of operational
SIAM J. Control,
21
(1964)
pp.54-59. [7]
Fuhrmann,
P.A.
[8]
Kirillova,
Notes on operators
F.M. and Chirakova,
and systems,
S.V.
Relative
to appear.
controllability
systems with time lag, Dok. Akad. Nauk. 176 [9]
Lions, J.L.
(1967)
erential A.
Ill] Ray, W.H.
and Lainiotis,
equations,
Springer
1971.
Optimal control of time lag systems with quadratic performance
indices,
D.G.
IV IFAC Congress,
(Eds), Identification,
and control of distributed Marcel-Dekker, D.L.
USSR,
pp.1260-1263.
Optimal control of systems described by partial diff-
[iO] Manitius,
[12] Russell,
of
Nonharmonic 18
systems,
1977.
Fourier series in the control theory
of distributed AppI.,
parameter
Warsaw 1969.
estimation
parameter
systems,
(1967), pp.542-559.
J. Math. Anal.
TABLE OF CONTENTS Page CHAPTER 1
1
INTRODUCTION Finite dimensional
linear systems theory
The quadratic control problem
CHAPTER 2
CHAPTER 4
2 3
The filtering problem
4
The stochastic quadratic control problem
5
Examples of infinite dimensional-systems
6 i0
SEMIGROUP THEORY Strongly continuous
CHAPTER 3
(regulator problem)
semigroups
iO
The infinitesimal generator of a semigroup
14
Hille-Yosida Theorem
17
Dual semigroups
25
Inhomogeneous differential equations
29
Analytic semigroups
33
Perturbation theory
37
Mild evolution oDerators
41
Quasi-evolution operators
43
Abstract evolution equations
44
CONTROLLABILITYF
OBSERVABILITY,
AND STABILITY
51
Exact controllability
51
Approximate
6O
controllability
Exact null controllability
68
Approximate null controllability
69
Observability
69
concepts
Duality theorem
71
Exponential stabilizability
72
QUADRATIC COST CONTROL PROBLEM
85
Regulator problem
85
Integral Riccati equation
88
Differential Riccati equation
93 96
Examples
102
Tracking problem Infinite time quadratic
cost control problem
107
Page CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
STOCHASTIC PROCESSES AND STOCHASTIC DIFFERENTIAL EQUATIONS
118
Abstract probability and estimation theory
118
Stochastic processes and stochastic integration
126
Stochastic evolution equations
143
Examples
149
THE STATE ESTIMATION P R O B L E M
158
•The filtering problem
158
Stability of the filter
170
Smoothing and prediction estimators
175
Differential forms for the estimators
180
Examples
183
Extensions
188
THE SEPARATION PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL
194
Admissible controls
194
Optimal control for complete observations
200
Separation principle for incomplete observations
204
Examples
207
UNBOUNDED CONTROL AND SENSING IN DISTRIBUTED SYSTEMS
212
Motivation
212
General conditions
216
Perturbation results
223
Controllability, observability, and stabilizability
230
The quadratic cost control problem
235
The estimation problem
244
TIME DEPENDENT SYSTEMS
252
Perturbation theory for mild evolution operators Examples
253 262
Abstract evolution equations
269
The quadratic cost control p r o b l e m
271
State estimation
278
Stochastic optimal control
282
Concluding remarks
283
ADDITIONAL REFERENCES
287
CHAPTER 1
INTRODUCTION
Most dynamical systems which describe processes in engineering, physics, and economics are extremely complex and the identification of mathematical models is difficult.
Consequently, early investigations of sys-
tems were confined mainly to analysing stability behaviour of very simple modelsusing
frequency domain methods.
In the last fifteen or so years
the state space approach has also become very popular and several new systems theory concepts such as controllability, observability, linear quadratic control, estimation and filtering, and realisation have been introduced and analysed
[i], [4], [5], [9], [iO].
However, these concepts
are only well understood for simple systems, namely linear difference equations and linear ordinary differential equations.
In the engineer-
ing jargon these simple systems are called lumped systems and they can be described by linear maps on finite dimensional linear vector spaces. For systems described by partial differential equations
(distributed
systems) or by delay equations the appropriate state space is an infinite dimensional function space and there has been some work on generalizing the systems theory concepts to special classes of these systems [2] ,[6],[7] ,[8]. Using a semigroup representation, we develop a self contained abstract theory for a wide class of linear systems, both finite and infinite dimensional which includes lumped, delay and distributed systems.
Results
are obtained which, when interpreted for a particular class of system, yields the known results.
Moreover, the abstract approach clarifies the
main ideas and mathematical problems so that new results are more easily obtained. We do not consider all the systems theory concepts here, but first concentrate on controllability, observability and stabilizability which turn out to be more complicated in infinite dimensions.
We then con-
sider the quadratic cost control problem and its dual, the filtering problem, and obtain the separation principle for infinite dimensional stochastic systems.
To motivate our approach we present a brief survey
of the finite dimensional theory concepts and results which we will generalize in Chapters 2 - 7.
Finite
dimensional
linear
Here we restrict ordin&ry loss o f the
ourselves
differential generality,
system
has
we
been
expressed
where
Zo,
rices
respectively,
(I.i)
is a d i f f e r e n t i a l
e R n, u(t)
theory systems
with that
by
in the + Bu
£ R m,
u is t h e
which
a given
are
a suitable
canonical
;
z(O)=
described
initial
state.
choice
by
linear
Without
of s t a t e
vector
form
z°
A a n d B are n × n and n x m r e a l m a t -
control
equation
term.
on the
state
space
R n and has
the
uni-
solution
(1.2) We
suppose
z = Az z(t)
to
equations
(i.i)
que
systems
z(t)
also
suppose
= eAtzo
that we have
(1.3) where
eA(t_s)Bu(s)ds
an a s s o c i a t e d
observation
of
(i.i)
y = Cz C is a r e a l
The for
+ ft o
k × n matrix,
following
(i.i)
and
Definition
(1.3)
i.i
The matrix constants
are n o w
(i.i)
such
implies
that
~Iz(t)ll ~ O as t ~ ~. ponential strictly
stability
and observability
standard. stability
is e x p o n e n t i a l l y
stable
if t h e r e
exists
positive
S
Me - ~ t
for the
for
uncontrolled
A necessary
and
of A is t h a t
the
all
t ~ O.
system
(i.I)
sufficient real
parts
(with u = O),
condition of
its
for
the ex-
eigenvalues
are
negative.
If one u s e s and this
controllability
that lleAtll
This
e R k.
of s t a b i l i t y ,
Exponential
A in M,~
concepts
so y
type
a feedback of control
control can be
u = -Fz,
used
then
(i.i)
to s t a b i l i z e
becomes
an u n s t a b l e
z=(A-BF)z system
= Az. Definition
1.2
(i.i)
or
that
A-BF
(A,B)
Another assigned
Stabilizability is s t a b i l i z a b l e
is e x p o n e n t i a l l y important
final
state
if t h e r e
exists
an m × n m a t r i x
F,
such
stable.
systems
theory
can b e
reached.
concept
is w h e t h e r
or n o t
a pre-
D e f i n i t i o n 1.3 (l.1) or
Controllability
(A,B) is c o n t r o l l a b l e
if any initial point z o can be s t e e r e d
to an a r b i t r a r y final point z in some finite time t I by some control u c L
(O,tl;Rm).
(A,B) is c o n t r o l l a b l e rank n :
[B : AB
: A2B
It h a p p e n s that if Controllability Definition We say that
if and only if the f o l l o w i n g n × nm m a t r i x has : ...
: An-IB].
(A,B) is c o n t r o l l a b l e ,
then
(A,B) is s t a b i l i z a b l e .
is also related to o b s e r v a b i l i t y .
1.4
Observability
(1.1),(1.3)
t ~ O implies Zo= O. k n o w l e d g e of y(t)
or
(A,C) is o b s e r v a b l e if
ceAtzo = 0
T h a t is for the c o n t r o l l e d s y s t e m
and u(t) on a finite time i n t e r v a l
for all
(1.1),(1.3),
a
[O,tl] u n i q u e l y
d e t e r m i n e s the initial state z O. (A,C) is o b s e r v a b l e if and only if the rank of the f o l l o w i n g n × kn m a t r i x is n :
[C': A'C':
... : (A')n-lc'].
So c o n t r o l l a b i l i t y and o b s e r v a b i l i t y are dual concepts that
(A,C) is o b s e r v a b l e
if and only if
(A',C')
in the sense
is controllable.
If the c o n t r o l l e r a n d / o r o b s e r v e r is d e s i g n e d so that the s y s t e m is stable and c o n t r o l l a b l e or o b s e r v a b l e ,
then the q u e s t i o n of o p t i m a l i t y
can be considered.
The q u a d r a t i c control p r o b l e m Consider
(1.4)
(regulator problem)
(i.i) and the cost f u n c t i o n a l
J(u)
= z(t I) 'Gz(t I) + f tl {z (t) 'Mz (t) + u ( t ) ' R u ( t ) } d t o
w h e n G, M and R are real s y m m e t r i c n × n, n x n, and m x m m a t r i c e s r e s p e c t i v e l y , w i t h G >_ O, M >_ O and R > O.
The r e g u l a t o r p r o b l e m is to
find an optimal control u* e L 2 ( O , t l ; R m) such that U n d e r the above assumptions, (1.5) w h e r e Q(t)
u*(t)
there exists
(1.4)
is m i n i m i z e d .
a unique o p t i m a l control
= -R-IB'Q(t)z(t)
is an n x n real s y m m e t r i c m a t r i x w h i c h is the unique s o l u t i o n
of the Riccati e q u a t i o n Q(t)
+ Q ( t ) A + A'Q(t)
+ M = Q(t)BR-IB'Q(t)
(1.6) Q(t I) = G The optimal cost is
J(u*)
= ZoQ(O)z o.
For G = O and tl= ~. we h a v e the i n f i n i t e time r e g u l a t o r p r o b l e m and if
(A,B)
and
(A',M ~) are s t a b i l i z a b l e ,
f e e d b a c k control of the form
there exists
a unique optimal
(1.5), w h e r e Q is time i n v a r i a n t and is the
u n i q u e s o l u t i o n of the a l g e b r a i c R i c c a t i e q u a t i o n (1.7)
QA + A ' Q + M = Q B R - I B ' Q
The f i l t e r i n g p r o b l e m We c o n s i d e r a noisy signal process
and a noisy o b s e r v a t i o n process
d e s c r i b e d by the f o l l o w i n g s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s (1.8)
dz(t)
= A z ( t ) d t + Ddw(t)
(1.9)
dy(t)
= C z ( t ) d t + Fdv(t)
;
z(O) = z o
w h e r e A, D, C and F are real n x n, n × m, k x n, k × k m a t r i c e s ,
z ° is
a G a u s s i a n zero m e a n v e c t o r r a n d o m v a r i a b l e w i t h c o v a r i a n c e m a t r i x Pop and w(t)
and v(t)
are i n d e p e n d e n t v e c t o r - v a l u e d W i e n e r p r o c e s s e s of
d i m e n s i o n s n and k and i n c r e m e n t a l ively.
The s o l u t i o n of
cess w i t h c o n t i n u o u s s a m p l e paths (i.iO)
c o v a r i a n c e m a t r i c e s w and v respect-
(1.8) is a zero m e a n G a u s s i a n s t o c h a s t i c proand is given by
z (t) = e A t z o + f t e A ( t _ S ) D d w ( s ) o
The f i l t e r i n g p r o b l e m is to find the b e s t e s t i m a t e ~ (t) of the signal process
z(t) b a s e d on the o b s e r v a t i o n s y(s), O _< s _< t.
The s o l u t i o n is the w e l l - k n o w n K a l m a n Bucy filter; ~(t) = E{z (t) I y(s); d~(t)
O <_ s <_ t}, w h i c h is given by
= {A-P(t)C' ( F V F ' ) - I c } ~ ( t ) d t + P(t)C' (FVF')-idy(t)
(1.11~
(o) = 0 w h e r e P(t)
is the u n i q u e s o l u t i o n of the R i c c a t i e q u a t i o n P(t)
= AP[t)
+ P(t)A'
+ D ' W D - P(t)C' (FVF')-IcP(t)
(1.12) P (O) = P P(t)
o
is the c o v a r i a n c e of the error process,
(1.13) If
P(t) = E{[z(t)(A' ,C') and
~(t)]
[z(t)- ~(t)]'}
(A,DW ½) are s t a b i l i z a b l e ,
in the sense that m e a s u r e s
i.e.
i n d u c e d by P(t)
then the f i l t e r is stable c o n v e r g e as tl+ ~ to P, the
unique s o l u t i o n of the a l g e b r a i c R i c c a t i e q u a t i o n (1.14)
PA'
+ AP + DWD'
= PC' (FVF')-Icp
The Riccati ator problem
equations
(1.12)
are equivalent
for filtering
if in
and
(1.6)
Q(tl-t) , D'WD by M, C by B', FVF' by R and Po by G. ution
shows
ationship
that
(1.14)
is known
and
(1.7)
for the regul-
(1.12) we replace A by A', P(t) A similar
are also of the same form.
as the duality between
the control
by
substitThis
rel-
and the filtering
problem. The smoothing
problem
and are concerned
and prediction
with estimating
problem
are also of interest
z(t) based on the observations
y(s),
O ~ s < tl, where t l > t for the smoothing problem and tl< t for the prediction problem. The best optimal predictor ~(tltl) satisfies the stochastic
differential
equation ~(tltl)
= A~(tltl)
CI.15) z(tlItl )= z(t I) where
~(tl)
smoother ~.16)
is the best optimal
is also given { dz(tltl)
filter up to time t I.
in terms of the optimal
= Az(tltl)dt
+ D'WDI(t)dt
filter
The best
linear
~(t)
+ P(t)C' (FVF')-Ic~(t)dt
(tllt I) = ~ ~t I) where I dl (t) = -(A-P(t)C(FVF,)-Ic)'I
(1.17)
(t)-C' (FVF')-l(dy(t)-C~(t)dt)
(t 1) = O and P(t)
is the covariance
The stochastic We consider (1.18)
quadratic
= Az(t)dt
with the noisy observation dy(t)
where we make filtering
usually strategy
+ Bu(t)dt
By admissible
and u(t)
we mean that y(s);
is to find the admissible
the separation
given by
principle,
is to use the deterministic
z (t) by its conditional
z(O) = z o
on A, D, W, Zo, C, F and V as for the
B is a real n x m matrix
problem
;
equation
+ Fdv(t)
value of the cost J(u)
termed
differential
+ Ddw(t)
only on the past observations,
astic control the expected
problem
process
= Cz(t)dt
control.
and depends
(1.12).
controlled
the same assumptions
problem.
stochastic
of
cost control
the stochastic
dz(t)
(1.19)
operator
expectation
feedback
O < s < t. control
(1.4).
because
The stoch-
which minimizes
The solution
the optimal
law of
~ (t), which
is an admissible
u e L 2 (~,P;L2(O,tl;Rm))
is
control
(1.5), replacing
is obtained
using the
Kalman Bucy filter results.
More precisely,
the optimal
control
strat-
egy is given by (1.20)
u*(t)
=-R-IB' Q (t) ~ (t)
I d~(t) = {A-P(t)C' (FVF') -I -BR-iB'Q(t)}~(t)dt (1.21)
+ P(t)C' (FVF')-idy(t) ~¢o)
where Q(t)
= o
and P(t)
The optimal
are the solutions
of
(1.6) and
(1.12)
TO motivate
the generalisation
ite dimensions,
we present
several
uted systems where questions ality might arise. Examples
respectively.
cost is then
trace{SP(t l) }+ ftltrace{MP (s) }as + Itltrace{Q(s)P(s)C' o o
Example
+
of linear infinite
(FVF')-IcP(s) }ds
of the above systems theory to infinsimple examples
of stability,
dimensional
of delay and distrib-
controllability,
and optim-
systems
i.i
Suppose we have a thin, narrow,
homogeneous,
continuous material
strip which is fed into a furnace by means of a variable-speed port mechanism. the diffusion
Then its temperature
zt(x,t)
= ~Zxx(X,t)
Zx(O,t)
= O = Zx(l,t)
+ v(t)Zx(X,t)
distribution,
usivity,~ is a constant proportional v is the material-strip
temperature
trans-
can be modelled by
equation
where z is the temperature material,
distribution
distribution
at some preassigned
temperature
~ is the coefficient
to the surface
velocity,
of the strip.
control u and that it is desirable
+ ~{z(x,t)-u(x,t)}
of diff-
conductivity
and u(x,t)
of the
is the external
We shall suppose that we can
to keep the outlet temperature
8(t) say.
z(l,t)
Thus we are led to the con-
trollability problem of whether the desired outlet temperature can be achieved and maintained. In general this will be impossible, and so instead we may seek to minimize
the functional
J =fotl{z (l,t)-8 (t)}2dt Usually the controls
are constrained
lu(t) l S 1 say, or by including the functional
J.
and we can express
a penalty
this by assuming
for using too much control in
Example
1.2
In steel m a k i n g plants
it is n e c e s s a r y to e s t i m a t e the t e m p e r a t u r e
d i s t r i b u t i o n of m e t a l slabs b a s e d on m e a s u r e m e n t s the surface.
A possible model
PClZt (x,t) = kZxx(X,t)
- e [ z ( x , t ) - Z o ( X , t ) ] + ~(x,t) Zx(O,t)
where
0, C l, k are the density,
;
0 < x < 1
= O = Zx(l,t) heat capacity,
c o n d u c t i v i t y of the m e t a l slab,
and e f f e c t i v e t h e r m a l
@ is a h e a t t r a n s f e r parameter,
the average coolant temperature, noise disturbance.
at certain points on
for the t e m p e r a t u r e d i s t r i b u t i o n is
and ~(x,t)
z O is
is some d i s t r i b u t e d w h i t e
The p r o b l e m is to e s t i m a t e the t e m p e r a t u r e p r o f i l e
z(x,t); O S x ~ i, t > O, b a s e d on the noisy m e a s u r e m e n t s Yi(t)
= Z(Xi,t)
+ ~i(t)
;
i=l,2,...,k
w h e r e xi, i = l , . . . , k are p o i n t s on the s u r f a c e of the slab and hi(t) r e p r e s e n t s the m e a s u r e m e n t error. Example
1.3
The e v o l u t i o n of the p o p u l a t i o n of a country can be d e s c r i b e d by the f o l l o w i n g linear h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n ~p(t,r) ~t
w h e r e p(t,r)
+
~p(t,r) Dr
--~(t,r)p(t,r)
p(O,r)
= Po(r)
;
O _< r < 1
p(t,O)
= u(t)
;
0 < t < tI
r e p r e s e n t s the p o p u l a t i o n d e n s i t y of i n d i v i d u a l s of age r
at time t, ~(t,r)
is the m o r t a l i t y function, Po(r)
age d i s t r i b u t i o n and u(t) trol variable.
is the given initial
is the b i r t h rate w h i c h we assume is the con-
The p r o b l e m is to choose u so as to achieve a d e s i r e d
age p r o f i l e q(r)
at the final time tl, and m a t h e m a t i c a l l y we could in-
t e r p r e t this as m i n i m i z i n g ii J(u)
=
tl
{P(tl,r)-q(r)}2dr + I o o
w h e r e the s e c o n d term m e a s u r e s Example
Xu2(s) ds
the social cost of c o n t r o l l i n g birthrate.
1.4
Suppose we have a s t r e t c h e d n o n u n i f o r m string w h o s e m o t i o n is desc r i b e d by
l
\
~(xlztt(x,t) - [~(x)zx(x,t) ) = v(x,t) x
z(O,t)
= O ;
z(l,t)
= u(t)
w h e r e z(x,t)
is the d i s p l a c e m e n t of the string at p o s i t i o n x, p(x)
the d e n s i t y d i s t r i b u t i o n of the string, meter,
u(x)
is
is a scaled t e n s i l e para-
and u is the c o n t r o l at the end of the string and v is a d i s t r i b -
uted control.
We may be c o n c e r n e d w i t h w h e t h e r we can choose u or v to
b r i n g the string to rest.
Alternatively,
we may w i s h to m i n i m i z e the
cost f u n c t i o n a l
J(u) =
[z a ( x , t ) + ) , l v 2 ( x , t ) ] d x d t + ~ , 2 f O
w h e r e X l, as
O
12 are p o s i t i v e
constants.
P h y s i c a l l y this can be i n t e r p r e t e d
k e e p i n g the s t r i n g as close to zero o v e r
as little control, Example
u ~(t)dt
0
[O,ti]×[O,13 , w h i l e u s i n g
as possible.
1.5
In the e c o n o m i c m o d e l l i n g of a trade cycle Kalecki m o d e l for f l u c t u a t i o n s of capital stock K(t) K(t) = aE(t) w h e r e a, b, c are constants,
+ bE(t-h)
[3] o b t a i n e d a
in the form + cu(t)
and the delay h r e p r e s e n t s the time lag
b e t w e e n the m o m e n t of d e c i s i o n to invest and the m o m e n t of d e l i v e r i e s of capital equipment.
The c o n t r o l p r o b l e m c o u l d be to o b t a i n a d e s i r e d
level of stock K d at a time tl, or to m a i n t a i n i n t e r v a l of time
[tl,tl+h ] .
Alternatively
ize the s y s t e m by a f e e d b a c k control, the f u n c t i o n a l
it at that level for an
it may be d e s i r e d to stabil-
or to r e g u l a t e the s y s t e m so that
ct 1 J =Jo
{(K(t)-Kd)2
+ lu2(t)}dt
is m i n i m i z e d .
Comments The above examples
are only a small t y p i c a l sample of p r o b l e m s w h i c h
arise in applications.
M a n y m o r e e x a m p l e s w i l l be found in r e f e r e n c e s
[2],[6],[7],[8]. In the next six c h a p t e r s we w i l l d e v e l o p an a b s t r a c t theory for time i n v a r i a n t i n f i n i t e d i m e n s i o n a l systems, w h i c h w i l l y i e l d s o l u t i o n s Examples
i.i, 1.2, E x a m p l e
are time invariant. 1.3 or to E x a m p l e
to
1.4 w i t h u = O, and 1.5 w h e n the c o e f f i c i e n t s
H o w e v e r this t h e o r y is not a p p l i c a b l e to E x a m p l e
1.4 w i t h u ~ O, w h e r e the control is i m p l e m e n t e d on
the b o u n d a r y of the d o m a i n of the system.
Control problems with
limit-
ed control action and e s t i m a t i o n p r o b l e m s w i t h limited sensing o c c u r f r e q u e n t l y in a p p l i c a t i o n s ,
so in C h a p t e r 8 we show how the theory can
be g e n e r a l i z e d to cover this i m p o r t a n t class of problems.
F i n a l l y in
Chapter
9 we generalize
pendent
systems
all the previous
theory to allow for time de-
and so all of the above examples
fall within the scope
of these lecture notes. While Chapters infinite trast,
2 to 7 are a self-contained,
dimensional
the style of Chapters
motivate
the abstract
detailed development
systems theory for time invariant 8 and 9 is expository.
formulation
quired for time-dependence
systems,
of
by con-
The aim here is to
of the extra technical
conditions
and limited sensing and control,
re-
and only an
outline of the proofs is given. References [i]
Brockett,
R.W.
[2]
Butkovskii,
Finite dimensional
A.G.
Theory of optimal meter systems,
A.
[3]
Kalecki,
[4]
Kalman,
A macrodynamic
[5]
Lee, E.B., Markus,
Econometrica,
equations,
Bibliography
[8]
Wang, P.K.C.
Control of distributed Press,
Springer,
systems,
systems;
in Advances
ed. by C.T. Leondes, systems, ii, Springer,
control;
in Economics
1974.
parameter
parameter
Sciences,
Linear multivariable Lecture Notes
by partial
1971.
i, 1964.
Linear multivariable Applied Math.
W.M.
1969. theory,
7 (1968).
in control systems, Academic
Springer,
of distributed
Int. J. Control,
[iO] Wonham,
control
1967.
Wang, P.K.C.
W.A.
in mathematical
McGraw-Hill,
of optimal
[7]
Wolovich,
cycles,
control of systems described
differential
[9]
Topics
theory,
L., Foundations
Optimal
para-
1969.
3 (1935).
R.E., Falb, P.L., Arbib, M.A.
J.L.
1970.
control of distributed
theory of business
Wiley, Lions,
Wiley,
American Elsevier,
systems
[6]
linear systems,
1974.
A geometric
and Math.
approach,
Systems,
iO1,
CHAPTER
SEMIGROUP
The 1 were
examples either
of i n f i n i t e
partial
2
THEORY
dimensional
or delay
systems
differential
o u r a i m is t o d e s c r i b e
an a b s t r a c t
formulation
(2.1)
z = Az
;
which
will
enable
dimensional Suppose
the
to p r e s e n t
z ° e Z is t h e space
dynamics
and
autonomous,
Tt,
such
then we
can
z(t) also
assume
Hadamard
Chapter
= zO treatment
zero of
state
at time
the evolution
for each
a)
of these
and
finite
time
a dynamical t is z(t).
If w e
from z O to
t define
system
z(t)
a linear
defin-
assume
are
linear
operator
state
(identity
o n Z)
of our dynamical
conditions,
system
satisfies
namely
it is u n i q u e it v a r i e s
continuously
z(t+s)
the dynamics
points
TO = I
= Ttz o
that the
the uniqueness,
by allowing
Z + Z ;
well-posed
b)
itial
in C h a p t e r
In t h i s
that
(2.3)
From
a unified
at time
govern
Tt:
the
state
Z, a n d t h e
which
(2.2)
L e t us
z(O)
introduced
systems.
ed on a Banach that
us
+ Bu
we
equations.
is t h e
with
the
same point
to evolve
f r o m z(s)
initial
state.
in Z as t h e p o i n t for a time
t,
reached
from all
in-
z O e z, h e n c e z(t+s)
= Tt+sZ o = Ttz(s)
= TtTsZ O
Tt+ s = TtT s Since we itial some
state
are a s s u m i n g Zo, w e k n o w
smoothness
t + O + for
all
on the
the
state
T t is a b o u n d e d trajectory
z o e Z, t h a t
2.1
A strongly
A strgngly
continuous
z(t),
as
continuous
semigroup
Tt+ s = TtT s
map
on
and
assume
Z, a n d
with
finally
that
z(t)
the we
in-
impose
÷ z o as
t + 0+ semigroup
is a m a p
isfies (2.4)
continuously
is
llTtZo - Zoll ~ 0 Definition
varies
0 ~ s ~ t
T t from R + to ~(Z),
which
sat-
11
(2.5)
T
(2.6)
=
o
I
l]TtZo - ZolI ~ O
Example
as
¥ zO E Z
t ~ O+
2.2
Let A £ ~(Z),
and set
(2.7)
eAt =
~ n=o
Clearly,
the right hand side converges
easy to verify conditions inuous semigroup, Example
(At)n n! in the uniform topology
(2.4),(2.5),(2.6),
and in fact is u n i f o r m l y
and it is
so e At is a strongly
cont-
continuous.
2.3
Let Z be the Banach space of continuous, with the sup-norm, (2.8)
and consider
(Ttz) (x) = z(x+t)
Clearly,
bounded
the t r a n s l a t i o n ;
(2.4), (2.5) are satisfied,
z £ Z,
and for
lJTtZo - ~oI~ : sup
functions
on
[O, =)
operator x >_ O
(2.6) we have
IZo(X+t)
- Zo(X) I
x>O + O as t ~ O + Example Let space,
2.4 {~n } n=l,2,..,
be an o r t h o n o r m a l
and {A n } n=l,2,..,
(2.9)
be a sequence
sequence
in a separable
of complex numbers,
Hilbert
then
xt e n ~n<~n,Z>
Ttz = n=l
is a b o u n d e d and this
linear o p e r a t o r
It if and only if {e n } is a b o u n d e d
is the case if sup Re{A n} < n
Under this assumption,
we have (t+s)
Tt+sZ = ~ e xn n=l and
~n<~n,Z>
Xn t TtTsZ =
e
0n<~n,TsZ>
n=l ~ =
n=l
Xnt e
~ ~n<~n ,
m=l
lmS e
~m<~m, z>>
sequence,
12
Z e n=l Clearly
(2.5)
I t I s n n e Cn<~n,Z>
is s a t i s f i e d ,
= Tt+sZ
and
I t llTtz - zll 2 = E le n _ i i 2 1 < ~ n , Z > i 2 n=l N i t I t = ~ le n _ l l Z l < + n , Z > l = + = le n _ l l = l < + n , Z > I = n=l n>N I t N sup Is n _ lla ~ l<~n,Z>[2 + K Z l<~n,Z>I 2 n=l, . . ,N n=l n>N for some K.
But
for N s u f f i c i e n t l y
large
and any e > O,
n>N H e n c e T t is a s t r o n g l y Some e l e m e n t a r y
continuous
properties
semigroup.
of s e m i g r o u p s
are c o n t a i n e d
in the f o l l o w -
ing t h e o r e m . Theorem
2.5
L e t T t be a s t r o n g l y (a) IITtlI is b o u n d e d (b) ¥ z e Z, Ttz
~0'
on e v e r y
iogllTt
B a constant
semigroup
finite
is s t r o n g l y
(c) if n O = t>oinf ~ (d) W n >
continuous
subinterval
of
space
Z, then
[O,~)
continuous
, then n o M
on a B a n a c h
~
t÷~lim
logllT t
s u c h t h at V t > O, llTtlI < M~e nt ---
Proof a) F i r s t w e s h o w t h a t origin,
that
IITtll is b o u n d e d
is 3 6 > O and M d e p e n d i n g
IITtll If t h is does n o t h o l d ,
! ~
on s o m e n e i g h b o u r h o o d
of the
on 6 such that
for t e [0,6]
then 3 a sequence
{tn},
t n ÷ O + s u c h that
lITt IIh n. H e n c e by the u n i f o r m B o u n d e d n e s s T h e o r e m 3 one z such that n {llTtnZll} is u n b o u n d e d , b u t t h i s is a c o n t r a d i c t i o n to the s t r o n g c o n tinuity
at the o r i g i n .
N o w set t = m6 + T, w i t h O ~ T ~ 6, then
IITtlI < ~ITJmIITTI( <_ M l + m <_ M M t/6 = Me et To p r o v e b) we h a v e
where
n = 6 - 1 1 o g M.
for f i x e d t > O, s > O,
llTt+s z - Ttzll <_ IITtlIIITsZ - zll <_ MentllTs z - zll
t3
Hence lim+}ITt+sZ-Ttzll
=
0
U
Also
for t > O, s ~ 0 s u f f i c i e n t l y IITt-sZ-Ttzll
Hence
lim
~
small
IIT t - s IIII Z - T s Z II
llTt+s z - zll = O, and Ttz
is c o n t i n u o u s .
s~ O-
c) L e t t
o n=l~2,...
> 0 be a f i x e d n u m b e r ,
then
for e v e r y t > t -
n t o -< t <
such that
log ILTtll
(n+l)t o .
log IITt T t _ n t =
o
Consequently II
o
t _<
n log IITtol I
log M
+
t <
t
log IITroll
nt °
t o
t
t~
+
<
o
t
t
< inf
o
l°gll Till
t>o
<
d)
If e > ~o'
t
lim
l°gll Ttll t
for
t > t
3 t o such that log llTtll < t
t h a t is
llTtll ~ e et
But
llTtlI ~ M O Me = Mo
e
-
for for
o b t a i n the r e s u l t .
o
t ~ tO
0 ! t ! to,
r e > O
M e = e-eto M O , and we
t
o
log ll~t II
li--~l°gll~tll
Thus
log M
o t
log M + t
log IITt II <
~ < 0
there exists o
so if
14
Example
2.6
It is easy to show that for the s e m i g r o u p s of E x a m p l e s 2.2, 2.3, sup Re~ t the bounds for IITtll are e ''Allt, i, e n respectively. Since we have only assumed Ttz is continuous, general to d i f f e r e n t i a t e Ttz.
However,
2.4,
it is not p o s s i b l e in
our m a i n aim is to relate Ttz
to the s o l u t i o n of an a b s t r a c t e v o l u t i o n equation,
and for this we need
the f o l l o w i n g concept D e f i n i t i o n 2.7
The i n f i n i t e s i m a l g e n e r a t o r of a s e m i g r o u p
The i n f i n i t e s i m a l g e n e r a t o r A of a s t r o n g l y continuous
s e m i g r o u p on a
B a n a c h space Z is d e f i n e d by i
(2.10)
Az
=
w h e n e v e r the limit exists; elements
lim ~ ( T t - I ) z t~o+ the d o m a i n of A, D(A) b e i n g the set of
in Z for w h i c h the limit exists.
E x a m p l e 2.8 The i n f i n i t e s i m a l g e n e r a t o r of the s e m i g r o u p of E x a m p l e 2.2 is d A e ~(Z). For Example 2.3 the i n f i n i t e s i m a l g e n e r a t o r is A = ~x ' w i t h D(A) = {z, ~
with
~ Z}.
For Example 2.4, the i n f i n i t e s i m a l g e n e r a t o r is
Az =
~. X n ~ n < ~ n , Z > n=l
D(A)
=
z :
Iln<~n,Z>l
<
T h e o r e m 2.9 Let T t be a s t r o n g l y c o n t i n u o u s s e m i g r o u p on a Banach space Z, w i t h i n f i n i t e s i m a l g e n e r a t o r A, then a) if z ° ~ D(A), d b) ~t(TtZo)
Ttz o e D(A)
= ATtz o = TtAZo,
¥ t _> O for z ° e D(A),
dn . (Ttz O) = A n T t z o = TtAnzo, z o e D(A n) dtn t d) Ttz O - z O = of T s A Z o d S , z O e D(A) c)
e) A is a c l o s e d linear operator, f) N D ( A n) n
is dense in Z.
and D(A)
t > O t > O
is dense in Z
15
Proof L e t s > 0 and c o n s i d e r T t + s Z o - Ttz °
(Ts-I)z O
Ts-I =--Ttz s
= Tt s
s
If z e D(A) the m i d d l e l i m i t e x i s t s as s ~ o e x i s t as well. In p a r t i c u l a r Ttz O e D(A), ative
of
Ttz o
equals
(Ts-I) Tt_ s
For
z
s
left derivative
exists
a nd is
TtAz o.
Part
c) f o l l o w s
on t h i s r e s u l t .
d) let Z* be the d u a l of Z a n d t a k e any
= S t d ~ < Z *, T u Z o > o where
limits
right deriv-
o
-s
H e n c e the s t r o n g
the o t h e r
a n d the s t r o n g
small
- TtZo =
by induction
O, and h e n c e
A T t z o = T t A z o-
For t > 0 and s sufficiently Tt-sZo
O
< , > denotes
the duality
pairing
t TtZo-Zo > = f
z* e Z*, t h e n
du
between
Z* and Z.
TuAZo> du
for
Hence
z o e D(A)
t = < z*,
f o
TuAZodU
>
Now t
t
t
Ts-I f muZ du = ~1 ~ T s + u z du _ ~1 f ~ z du s Letting
o
o
p = s+u in t h e f i r s t
t 1 Ts-Is o~ Tuz d u = s !
integral,
u we have
1 t Tpz d~ - s of TuZ du = ° L o T p z d p
t+s +It Tpz dp
]
<,u } 1
s
= ~ /Tu(Tt-llz
du
o N o w as s ÷ 0 + t h e continuous.
~2.11)
right
hand side
tends
Hence t f TuZ d u o
~
D(A)
to
(Tt-I)z
since
Ttz
is
strongly
18
t
But
los
D(A)
which
TuZdU
tends
To p r o v e z such so
* z
as
t ~ 0 +, h e n c e
to z.
This
A is c l o s e d
we
t h a t Az n c o n v e r g e s
T s A Z n ~ TsY
shows
let
D(Ai
{z n} b e
to y.
uniformly
for
on
any
z ~ Z,
B a sequence
in
= Z.
a sequence
in D(A)
converging
T h e n llTsAZn-TsY N ~ M e ~s llAZn-yll
~O,t~.
Now
since
Zn ~ D(A),
to
and
we have
t Ttz n -
Using
the
Lebesque
zn
= f TsAZndS o
dominated
convergence
theorem
t Ttz
-
z = f
TsY ds
O
Thus
z ~ D(A)
and
Az
= y
, which
proves
A is closed.
+ For
part
f)
on R + h a v i n g ort.
Then
~(U)TuZ the
let
)
be t h e
derivatives
if ~ £ C ~ ( R +)
so does
class
~r,
vector
valued
set o f all e l e m e n t s
of the
form
g = f~(U)TuZ o
We w i l l
show
that
of
of all
is a c o n t i n u o u s
(2.12)
For
Co(R
continuous
du
the
r-th
~s -I s
real
valued
and of
functions
compact
derivative
function
of #,
on R + to
Z.
supp-
and
Let
Z ° be
z c Z, ¢ e C o ( R + )
Z e D ( A r) ¥ r = l , 2 , . . , o small
s sufficiently
all
orders
and
Z
is d e n s e
o
in Z.
1 f~ (u)[T u + s Z - T u Z ] du
g = ~
O ocl
= s 1 f {@(U-s)
- @(U)}TuZ
du
o (u-s)
But Thus
we
-
~(u)
-~'
s
Ag = -
see
that
/ ~5, ( U ) T u Z o
as
du
s ~ O+
.
Hence
uniformly g
e D(A).
with
respect
Repeating
this
to u. argument
g e D ( A r) ¥ r > O, a n d ao
Arg
=
(-l) r f ~ r ( u ) T u Z
du
O
which
shows
that
Z
c o
Suppose Banach
now
the
theorem
63 D (A r) r=l
closure a bounded
= 0
of
Z ° is n o t
linear
Z, t h e n
functional
3 z ° e Z and by t h e
z* on o
¥ g c Z ° and < z ~ , Z o >
Z such
= 1
that
Hahn-
17
Thus
<~,
f'~(U)Tu~ du
>
=
{~(u)
o
< z~,TuZ
¥ ¢ ~ C~(R +) and z e Z.
But
Hence
3 ¢ c C~(R +) such t h a t - f ~ ( u ) o c o n t r a d i c t i o n so ~o = Z.
< z~,TuZo>
the solution of an abstract = Az,
z(O) = z
du ~ 0, and this is a
play an important
evolution
continuous
semigroup
obtain a c h a r a c t e r i s a t i o n continuous
semigroups.
Theorem 2.10 A necessary
and z O g D(A).
This is p r o v i d e d by the H i l l e - Y o s i d a
semigroup
and sufficient
is that
condition
3 real numbers
Theorem.
for a closed linear operator A
Z to generate
M,~,
llR(l,A)rll
<
a strongly
continuous
such that ¥ real I > ~, I £ ~(A),
M
-
=
(~_~)
r = 1,2 .... r
(~I-A) -I is the resolvent.
(2.14) Proof
strongly
set of A, and
(2.13)
where R(I,A)
a to
Theorem
with domain dense in a Banach space the resolvent
if A generates
So it is important
of those operators w h i c h g e n e r a t e
Hille-Yosida
role in
equation
o
In p a r t i c u l a r we know z(t) = Ttz o is the solution strongly
= 0
is continuous with = 1.
The above t h e o r e m shows that semigroups determining
> du
o
}ITt II ~
In this case
Me ~t
We need the following Lemma
Lemma 2.11 Let T t be a strongly
continuous
A, and llTtll ~ Me ~t.
If Re ~ > ~ then ~ ~ p(A), and R(I,A)z
=
semigroup with
f~e-ltTtz o
infinitesimal
generator
dt
Proof Let
Rlz = _ #z e -~l t T t o
This o p e r a t o r
dt
is well defined,
z e Z, Re ~ > since
lle-ktTtzll ~ M e ( ~ - ~ ) t IIzll M o r e o v e r the i n t e g r a n d
is s t r o n g l y
where
~ = Re {~}
continuous,
hence strongly m e a s u r a b l e
18
and so the i n t e g r a l IIRxll _< We now s h o w Rlz
is a w e l l
defined
Bochner
M~ e - [ ~ - ~ ) t dt - ~-~M o
£ D(A)
and
T -I
i
_(II-A)RIz
integral.
We h a v e
so R x is b o u n d e d .
= z W z e Z
~e-lt
I
o
elS-ls f ~ e - l t T t z --~
-
-sl f S e - l t T t z dt
s
Thus
ARAZ
Also
RAAZ
--
lira s~o÷ k/ -T s~-)I ~ ~xz
o
= ~Az-~'
~ z ~
~"
o0
= / e -At TtAz o
dt = A/ e -It Ttz dt = A R A z o
Hence
RI(II-A)z
= z ,
z e D(A)
and
(AI-A)RIz
= z ,
z £ Z
so
R(I,A)
,
z e D(A)
= R1
Necessity By the
l e m m a we k n o w
~ e p(A)
if
Re I > ~, and
co
R(A,A)Z
= / e -At Ttz dt o
,
z ~ Z,
Re I > c0
oo
Hence
d(r-l)R(IrA)Zdxr_ 1 = R (r-l) (l,A)z
= of (-t) r - l e - A t
Ttz dt
r=l, 2,.. • co
ThUs But
fIR (r-l) (A,A)I~ _< M f t r - l e - ( ~ - ~ ) t o the
resolvent
is
R (r-l)
analytic
(A,A)
Hence
in
=
A e p (A),
dt = M(r-l): (if-m) -r
so
(-I) r-I (r-ll,(R(l~%)) r
(~_~) r
Sufficiency Set
A A = A2R(A,A)-AI
, A > ~, then
A 1 e ~(Z),
and w e may
construct
the s e m i g r o u p A = eAAt = e - A t Tt We w i l l
show that
the s t r o n g
~X~)
(AI_A)-n
n=o
limit
of T tI e x i s t s
as A ÷ ~ and is the
19 desired
semigroup
T t.
First we show
IIA~--A-II ÷ O as ~ ~ - , z ~ D ( A )
If Z ~ D(A), we have
II ~ ( I I - A ) - l z
-
_<
Ilt(~-A)-lll
Now
z
XM ~_~
<
2M
Hence we may apply the Banach-Steinhaus (AI-A)-Iz Now
Alz = I(II-A)-IAz,
II <_ ~
II = II(~I-A)-IAz
+ z
as
M
IIAz II
for large 1 theorem
I ÷ =
to conclude
that
for all z e Z.
so that Alz ~ Az as I ÷ ~
for z e D(A).
~I~ ~t Note that
llTltll <
e-lt ~
-
Now
n=o
(II-A)-I(~I-A)-I
A ~ I ~ -- T~A,~
(12t) n
=
, so that
n'
M
= M
e ~--~j
(I-~) n
(~I-A)-I(II-A) -I, hence
AIA ~ = A AI,
and
if ~ ~ D ( A )
t - T2
o
S
d U ~s (Tt-sTs z)ds
o
=
/
t
T~_ s(AI-A
)TlsZ ds
o t
=
TUt-sTsl (A I _ A )z ds
f o
t
-(l-U)m2s
Thus
iITt~z-T~zll < M2e~---~t II(AI-A~)z II ~ exp o
Choosing
I > ~, we have
llT z BUt
II (AA-A~)z{I ÷ O
Thus
T~z
converges
as
Now D(A)
theorem we may extend
ds
ll(A A )zllt
I,~ ÷ -, since
to a limit,
we will denote by Ttz. Steinhaus
t
T zll M 2 e
(~_~)(X-~)
uniformly is dense
Alz_ ÷ Az on compact
I ÷ -.
intervals,
which
in Z, so using the Banach-
this convergence
mains to show that T t is a strongly
as
continuous
to each z e Z.
semigroup
whose
It regenerator
is A. We have Tt+sZ
= lim Ttl+sZ = lim TtXTksZ = T tTsZ
Also To= I, and the strong convergence
on compact
continuity
intervals.
¥ z ~ S
is a consequence
of the uniform
20
Now
IIT~tAAz - TtAZ II <-
Hence
T~tAlz
converges
pact intervals, convergence
IIT~tIIIIAA Z - Azll +
s t r o n g l y to
for z e D(A).
ll(T~t - Tt)Az n
TtAz
as
~ ~ ~ , uniformly
Thus we may apply the Lebesque
on com-
dominated
t h e o r e m to t -
=
TsAxZ
ds
o obtaining
t Ttz - z
So the g e n e r a t o r ~
=
f o
TsAZ ds
z ~ D(A)
,
of T t is an extension
of A, since
Ttz - z ~z = lim t+o+
t
= Az
for
z e D(A)
Now if ~ >
and But
AD(A)
(II-A)D(A)
= Z
(~I-~)D(~)
= Z
= ~ D(A) ,
hence
(II-A)D(A) So that
D(A)
In Example
= D(~)
=
(lI-~)O(~)
and this completes
2.4 we defined a s e m i g r o u p
on a separable
Hilbert space,
and derived its g e n e r a t o r
in Example
ation of the H i l l e - Y o s i d a
T h e o r e m by p r o v i n g the converse.
Example Let
2.8.
the proof of the theorem.
We now illustrate
an applic-
2.12 {I n} be a sequence
sequence
in a separable
of complex numbers,
Hilbert
space H.
and {~n } an o r t h o n o r m a l
Now define an o p e r a t o r
on H
by
with
AZ =
~ ln~n<~n,Z> n~l
D(A)
= {z :
n~__l,in < #n,Z> 2
Then A is a densely d e f i n e d closed operator, if and only if uous semigroup
infll'l I > O. n n if sup Re{l n} < n
< and
} (II-A) is invertible
M o r e o v e r A generates
a strongly
contin-
21
Proof Clearly lle in D(A)
all sequences
{Zp} with
= O for n s u f f i c i e n t l y
large
and form a dense set in H.
Now let {Zp} be a sequence Since the sequence
{AZp}
£ Thus and so
<#n,Zp>
in D(A) with Zp÷ z ° and A Z p + Yo as p + ~.
is b o u n d e d we have
In<¢n,Zp> I
_<
M
£ ]In<~n,Zo> I z -< n=l
S
for p=l,2,...
A Z o = Yo' showing that A is closed.
C o n s i d e r now the e q u a t i o n (2.15)
lz - Az = y ,
If we set
y = n~=lSn<¢n,y>
then
(2.15)
is e q u i v a l e n t
z e D(A), y g H
to
(l-ln)<~n,Z>
= <#n,y>
Hence
<~n,Z> = l-~ l <¢n,y > n
for
X ~ 1n
and
n~=l~n<~n,Z>
if and only if
co
= z ~ H
infll-lnl
=
Thus
(II-A)
is invertible
From the H i l l e - Y o s i d a if there exists
if this condition holds
and
constants
_<
r=l,2
M
(l-~) r
~ n=l
IIR(x,A)rll <
....
¥
1 Cn<¢n,y > (l-ln) r
sup n
1 ]l_lnlr
So we need to prove that there exists M,~ such that
/ (2.16)
X-~ sup ~
tn
a semigroup
M,~ such that
R(I,A)ry =
so
and ~ e p (A).
T h e o r e m we know that A generates
]IR(X,A)rl[ Now
> O.
n
\k <
M
real
t
>
22 But
if Re{l n } ~ ~
, we h a v e <
iX_An I and h e n c e the one
(2.16)
given
in E x a m p l e
In g e n e r a l Theorem, eloped,
and one
Definition L e t A be on
it is d i f f i c u l t so o t h e r
2.13
The
a closed,
more
D(A*)
Z.
the
semigroup
generated
by A
is
of
conditions
verifiable
of the
conditions
of the
adjoint
Hille-Yosida
have
been
dev-
operator.
operator
defined, adjoint
all
the
linear
A*
those
operator
o f A is
z*
with
domain
a transformation
c Z* f o r w h i c h
there
D(A)
whose
exists
a
that
In t h i s
case we
Theorem
2.14
L e t A be
easily
adjoint
The
consists
to c h e c k
is in t e r m s
densely
space
e Z* s u c h
Clearly
2.3.
in p a r t i c u l a r
a Banach
domain g*
is s a t i s f i e d .
1
-
set
A'z*
a closed,
Z, t h e n A g e n e r a t e s t ~ 0 if and o n l y
= = g*.
densely
defined,
a semigroup if for
¥ z e D(A)
all
T t on
linear
operator
Z satisfying
on a B a n a c h
lITtli ~ e ~t f o r
space all
I >
(2.1~)
II(AI-A)zll z
~
(2.18)
II(AI-A*)z*IIz,
(A-~)llzll z , ~
z ~ D(A)
(A-~)IIz*IIZ, , z*e D(A*)
Proof Sufficiency First case
then
we
show
that
for s o m e
(AI-A)
Az - Az But
from
(2.17)
vertible.
Its
Yn = lZn-AZn'
we
to o n e
onto
Z.
If t h i s
is n o t
implies
is a c l o s e d z n e D(A)
I[Yn-YmIIZ
z = O so w e k n o w
subspace
of
Z since
a n d Y n ~ y as n + ~,
that
(hi-A)
if w e
is in-
set
then
= ilX (Zn-Z m) - A (Zn-Z m) II (l-e) llZn-Zmll
Thus
z n is a C a u c h y
Moreover, range
of
since
(II~A)
s e q u e n c e r and (AI-A)
is n o t
the
we have
= O
see this
range where
is o n e
z E D(A),
so z n + z o as n ÷ ~.
is c l o s e d ,
the whole
z o ~ D(A)
space,
then
and y = Az o. there
exists
If t h e z*
e Z*,
23
z*@ O,
such
that < z * , ( ~ I - A ) z>
In w h i c h
case
l
from
a n d so
z* E D ( A * ) ,
and
= ~z*
(2.18)
ll(x~-A*)z*11z* SO z* = O, Now
¥ z E D(A)
= ,
A'z* But
= O
from
thus
II-A
is o n t o
~ (x-~)llz*llz. Z.
(2.17) 1
Hence
i il(lI-A)-rll -<
and we m a y
apply
a strongly
continuous
the
r : l , 2 ....
(~_~yr
Hille-Yosida
Theorem
to c o n c l u d e
that A generates
semigroup.
Necessity If A g e n e r a t e s then by
the
a strongly
Hille-Yosida
continuous
semigroup
T t w i t h ilTtli ~ e ~t,
Theorem
fIR (I,A) II
1 ~_~
~
for
i >
or e q u i v a l e n t l y ilIZ - Azli _> (l-~)llzll Moreover,
if z*
for
z c D(A).
E D(A*)
ll~z* - A*z*ll = l[z sup 1<(~-A*)z*,z>l I[. llzli~I zED (A) sup _ li -ilR (l ,A) yll. -
sup l[ II~II ~
(~-~)~
= (~-~) llz*ll F o r the ditions
or
special
case where
Z is a H i l b e r t
(2.17), (2.18)
in a s i m p l e r
<~z-Az,
>
~z-Az>
form;
(~-~)2
2l(0~l[zl[ 2 - R e < A z , z > )
+
space
we
is
(2.17)
that for
I > ~
- ~211zi12 _>
can
,
rewrite
is e q u i v a l e n t z E D(A)
O
conto
24
The
above
a ~ such
inequality
holds
~llzll2 Similarly
(2.18)
will
Z
be
Finally
we note
that
restriction
T t on
Z with
alent
norm
for ~ if t h e r e
exists
if w e
it is e a s y it s h o u l d
to
to t h e
Example
2.15
and
that
given
llTt~] ~ e ~t
a Banach
possible
is n o t
space
Z and
to i n t r o d u c e
an e s s a semigrou~
an e q u i v -
that ~
emtI[ z II,
check
=
sup
that that
new norm
this
D(A) it is e a s y
= {z
to s h o w
A*z
e-~tllTtzll; norm
if the
I] II,
Az = - dz d-~ '
Let
z e D(A*)
set
be n o t e d
respect
are
it is a l w a y s
Z such
z e D(A)
Re{}
if we
llTtI[ ~ Me et
llzll.
Then
choice
if
the h y p o t h e s i s
since
II II, on
Re{}
~
HTtz ]], In fact
a suitable
satisfied
811zll2
ential
with
that
has
the
original
, this
t Z 0 required
space
property
properties.
is a H i l b e r t need
However
space,
with
n o t be
retained.
T t on
Z and
lITt II < 1 .
~(0)
= zI
Z = L 2 [O, I]
:
z e HI[O,I],
z(O)=
0 }
that
dz d~
=
Moreover =-%z
2(I)
< ~Jlzll2
= - % z 2 (O) <_ S[Izl[= where
~ = O.
Example
Hence
A generates
2.16
Consider
the
system + ~ ~ + Az = O
where with
A is a p o s i t i v e domain
D(A),
z = y
self
adjoint
>
kllzll2
, then =
,
z(O)
= z o,
operator
on a
~ z e D(A)
,
real
so t h a t
Let
a semigroup
-~ y - Az
k > 0
, ~ _> O Hilbert
space
H,
25 We may w r i t e this
formally as a first o r d e r s y s t e m --Aw
where
w =
[;]
, and
[o
I n t r o d u c i n g a H i l b e r t space ~ C = <w,w~
=
D(A½)×
H
H +
w i t h inner p r o d u c t H
w rw [;]0 [ ]we ve <W,~W~
=
+
= -~llyll 2
for w e D(~)
= D(A)
The a d j o i n t o f ~
× D(A ½). w i t h r e s p e c t to the H i l b e r t space ~ C
to be
[: ][:] _
Thus
DCA')
= DCA)
<w,J[~w> = -~[ly[l2, and so we may apply T h e o r e m 2.14 to
conclude that ~
2.1
,
is easily shown
generates
a s t r o n g l y c o n t i n u o u s s e m i g r o u p on ~ .
Dual s e m i g r o u p s First we recall the f o l l o w i n g p r o p o s i t i o n
from
[ii].
P r o p o s i t i o n 2.17 Let A be a linear o p e r a t o r w i t h D(A) a) The dual A* is a weak* A is bounded,
then A*
closed
E ~(Z*),
and if in a d d i t i o n
and IIA*[I = flAIl.
b) If A is c l o s e d then D(A*) flexive D ~ * )
dense in Z, a Banach space. linear operator,
is w e a k * dense in Z*, and if Z is r e -
is s t r o n g l y dense in Z*.
A s t r a i g h t f o r w a r d a p p l i c a t i o n of this p r o p o s i t i o n t o g e t h e r w i t h the Definitions
2.1 and 2.7 leads to the f o l l o w i n g theorem.
T h e o r e m 2.18 Let T t be a s t r o n g l y c o n t i n u o u s s e m i g r o u p on a B a n a c h space Z, then T~ is a linear o p e r a t o r w i t h ilT~ll = llTtll, and a} b)
T* = I* (identity on Z*) o T ' t + s = T~T*~ s ' t,s _> 0
C)
weak*
lim T~z* = z* t+o +
¥ z* e Z*
28 In general T~ will not be strongly mainly weak*
interested topologies
in r e f l e x i v e
continuous.
Banach spaces
are equivalent,
However we will be
for which the weak and
and so T~ will be w e a k l y continuous.
T h e o r e m 2.19 If T t is a w e a k l y continuous is also strongly
semigroup on a Banach space Z then it
continuous.
Proof Let z(t)= Ttz o, then z is w e a k l y s t r o n g l y measurable,
[5]) thus z is B o c h n e r Now let
continuous
¥ t > O.
Hence z is
and [~z (t)l~ is b o u n d e d on compact intervals
(see
integrable.
O <_ e < q < 8 < ~-£ < ~, e > O, and set z(~) = Tsz o = T~T~_~z o = T~z (~-~)
8
B
Then
(8-e)z(~)
= f z(~)dn = f Tqz(~-H)d~
SO
(S-m){z($+£)-z(~)}
= ] Tq{z(~+-e-~)-z(~-~)}d~
and
(8-~)IIz(~+-c)-z(~)U
<
Now
f
B $-s sup IITDII£_8 llz(s+e)-z(s)llds
~-s IIz(s+-~)-z(s)l]ds
+ o
To see this we note that the result functions
z, and for z B o c h n e r
as
c ÷ o
is certainly
integrable
true for continuous
B a continuous
function z,
such that for any e' > O $-~
Hence
f
II,(s)-~(s)lld~
]
l~z (s±e)-z (s)IIds
<
~,
~
Iz (s±E) ~ (s±e)ll
+11~(s±c)-~ This shows that z(t)
is strongly
Now let t n be any p o s i t i v e
ds
for t > O.
rational n u m b e r with t n < i, and consider
the countable
set S of all finite
a n a rational
scalar.
then S is w e a k l y
continuous
(s)l[ +11~(s)-z (s)ll]
linear combinations
ZanTtnZ °
If we denote the strong closure of
with S by S,
closed and since TtnZ o ~ z o as t n + O in the weak
27
topology
it f o l l o w s
t > O, w e h a v e
t h a t z o e S.
TtTtnZo
F r o m t he s t r o n g
= Tt+tnZo,
T t T t n z° ÷ T t n Z o
as
continuity
of T t for
a n d so
t + O
F o r any z m e S zm = Z a n T t
zo n
where
the s u m is finite, Ttz m ÷ z m
Hence
a n d so as
t ÷ O
.
H TtZo-Zoli < llTtZm-Zmil + UZm-Zoil + ~iTt(Zo-Zm) u
<,TtZmZmll +[sup ,Tt,÷l],Zm o, tt~ [o, l] But
inf ilZm-Zoli = O z e S n
as t + O,
Theorem
and this
and
Ttz m
completes
semigroup
T t is a c l o s e d ,
Banach
space,
if z* c D ( A * ) ,
then
T~z*
converges
c D(A*)
t = f T *u A * z * o
z* £ Z* b e l o n g s
in t h e w e a k *
A of the s t r o n g l y
defined,
If z* e D ( A * ) ,
topology
Hence
T~z*
e D(A*)
then =
du
to D(A*)
if a n d o n l y if
as s + O +, and the w e a k *
for e a c h
and
if Z
with
T*-I* s z* s l i m i t is A'z*.
Proof a)
cont-
linear operator
= T~A*z*
T~z*-z* an e l e m e n t
generator
densely
and
A*T~z*
b)
Ttz ° ÷ z o
2.20
is a r e f l e x i v e a)
t ÷ O, and so
the proof.
The d u a l A* of the i n f i n i t e s i m a l inuous
as
m
A*T~z*
f i x e d t > O, and all z e D(A) =
= TEA*z*
=
28
t [ du o
Moreover
t = f du o t f TuZ o
=
Now we
showed
in the p r o o f
of T h e o r e m
2.9
du>
=
t Af TuZ o
du>
that
t
Af TuZ du o t Hence
topology
a s s + 0 +. =
= z*
du
z* e Z* is s u c h
Hence
Ttz-z
f o
b) A s s u m e weak*
=
£ D(A*)
Conversely,
and
z c Z
T*-I* ~
that
Then
s
z*
which
converges
proves
to g*
in the
T*-I* s s
Z*,Z >
=
lim < z*, s~o +
Ts-__~Iz > s
= g*
fixed
sl
z*
£ D(A*)
=
du o
Since
1 s
TuZ
du + z
in t h e
=
1 s 5 f o
=
strong
lim s~o +
s 1 s f TuZ o
topology
=
du
du >
as s +
O+
%;
we h a v e
z £ Z
Remark The
dual
operator
A*
is the
a).
¥ z e D(A)
lim < s~o +
A'z*
for a n y
=
~
infinitesimal
generator
of T~
i
29
2.2
Inhomogeneous We h a v e
seen that
t h e n the s o l u t i o n
ks
z(t)
where
if A g e n e r a t e s
,
z (O) = z O
z(O)
for the m o m e n t w e w i l l on
,
continuous
semigroup
Tt,
z O e D(A)
the inhomogeneous
z = Az + f ,
a solution
a strongly
= Ttz O
let us c o n s i d e r
(2.19)
equations
of
= Az
Now
differential
= z
equation
o
assume
f £ C(O,tl;Z).
Suppose
that
z is
[O,tl] , t h e n
d-s
t-s z(s)
= -ATt-sZ(S)
+ Tt-s
z(s)+f(s
-- T t _ s f (s) Hence (2.20)
z(t)
It m a y be t h o u g h t
t = Ttz O + ~o T t - s f ( S ) that
is not t r u e in g e n e r a l . partial
converse.
Theorem
2.21
If A g e n e r a t e s
(2.20)
ds
is a l w a y s
a solution
of
H o w e v e r w e are a b l e to p r o v e
a strongly
continuous
semigroup
(2.19) b u t t h i s the
following
T t on a B a n a c h
space
Z, and a)
f e C I (O,tl;Z)
b)
z
then of
o
c D (A)
(2.20)
is c o n t i n u o u s l y
differentiable
on
[O,tl]
and is a s o l u t i o n
(2.19).
Proof Clearly
all we n e e d to s h o w is t h a t
differential
equation
(2.17).
t Let
v(t)
= f Tt_sf(s)ds o
t f Tt_sf(s)ds o
satisfies
the
30
then
v(t)
= of T t _ s
(O) + ~ f' (u)d~ o
t
t t
= I Tt_sf(O) as +
f I Tt_sf'(~as
o Now
Tt_ez
H e n c e v e D(A)
ds
da
o
t - z = A I Tt_sZ @
ds
for all
z e Z
and t Av(t)
=
(Tt-I)f(O)
+ I o
(Tt_ -l)f' (e)de
t = I Tsf(t-s)ds o
Now
v(t)
Thus
atd-Xv =
t
Ttf(O) + ~ Ts f' (t-s)ds o
Therefore dv d--~ = The conditions where
of Theorem
in g e n e r a l
content with weaken
Lemma
Av(t)
+ f(t) 2.17 are t o o s t r o n g
we do not wish
solutions
our hypothesis
to a s s u m e
satisfying
(2.17)
for c o n t r o l
applications
f e CI (O,tl;Z). almost everywhere,
If we are w e can
on f.
2.22
If A g e n e r a t e s
a strongly
continuous
Z and Tt_sf(S)
c D(A)
and
e LI (O,t; Z), t h e n
ATt_sf(S)
for a l m o s t
all
semigroup
T t on a B a n a c h
space
t > s e [O,tl] , f ~ LI (O,tl;Z)
(2.20)
is the u n i q u e
solution
of
(2.19). Proof Let
v(t)
assumptions,
t = ] T t _ s f ( S ) d s , t h e n s i n c e A is closed, o v(t) £ D(A) a n d
Av(t) Hence
Av(t)
under
the a b o v e
t = ; ATt_sf(s)ds o
is B o c h n e r
integrable,
and
e t
f Av(t)dt -- f f ATt_sf(s)ds at o
o o = I f ATt-sf(s)dt o s
ds
by Fubini's
theorem
31
= f[Tu_sf(S)-f(s)]ds o
=
Hence
v(a)
is differentiable everywhere. lications
v(~)
-
almost
2.9(d)
e D(A)
for t-s > O
~ f(s)ds o
= ~ f(s)ds o
However
by Theorem
since Tt_sf(S)
+ ~ Av(s)ds o
everywhere
and so satisfies
even this result
so instead we choose
(2.17)
is too restrictive
to work with
almost
for most app-
the input-output
relation
(2.20). Definition If
2.23
Mild solution
f ~ Lp(O,tl;Z) , p ~ i, we say that
(2.21) is a mild solution We note that
of
(2.19)
(2.21)
on
[O,t~
i
is well defined when the integral
in the sense of Bochner Lemma
t = Ttz O + f Tt_sf(s)ds o
z(t)
is interpreted
and we show that
2.24
z(t) defined
by
(2.21)
is strongly
continuous
on
[0, tl] •
Proof Without
loss of generality
we can assume
z o = O.
t = f (Tt+~_s-Tt_s)f(s)ds O
z(t+~)-Z(t) Then
t+6 + f Tt+~_sf(S)dS t q1
t+~ llz(t+6)-z(t)II
<_ I I ( T g - I ) z ( t ) I I +
I I T t + s _ s l l q ds
+ 0
(2.6)
as ~ + 0 + by
For ~>O, consider
and Theorem
51
P
I t
II f ( s ) ~
d
2.5(b)
Now consider z (t-6)-z (t)
=f
t f Tt_sf(S) ds t-~
t-~ (Tt_~_s-Tt_s)f(s)ds-
o Then
t-~ llz(t-~)-z(t)ll
Now
(Tt_~_s-Tt_s)f(s)
integrable
t
~ Of l~(Tt-~-s-Tt's)f(s)~Ids ÷ O as ~ ÷ O
from Theorem
2.5(a)
and
and since
+ ~-~ ~ITt-sf(s)llds
U(Tt_~_s-Tt_s)f(s)II f e
LP(O,tl;Z ).
is Bochner
32
So the first t e r m c o n v e r g e s to zero as ~ ~ O by the L e b e s q u e d o m i n a t e d c o n v e r g e n c e theorem.
The s e c o n d t e r m tends to zero by T h e o r e m 2.5(d).
We can show that this m i l d s o l u t i o n is the same as the concept of w e a k s o l u t i o n u s e d in the study of p a r t i a l d i f f e r e n t i a l equations.
D e f i n i t i o n 2.25 and f e LP[O,tl;ZJ,~" then we say that
Let Z be a r e f l e x i v e B a n a c h space, z is a w e a k s o l u t i o n of (2.22) if
z = Az + f ,
a)
t ÷ z(t)
b)
f o
for all
z (O) = z
is c o n t i n u o u s on
tI
o
[O,tl]
t1 < g ( t ) , z ( t ) > at + f o
< x ( t ) , f ( t ) > dt
+ <x(O),Zo>
= 0
g £ C[O,tl;Z* ] , where t1 x(t)
Proposition
=-
f t
T*s_tg(s)ds
2.26
For e v e r y
z° £ Z
there exists a unique w e a k s o l u t i o n of
and this is the m i l d s o l u t i o n of
(2.22)
(2.22).
Proof Substituting
for x(t) we see that
b) is e q u i v a l e n t to
f tl > dt o
=
O
w h i c h shows that the m i l d s o l u t i o n is a w e a k s o l u t i o n and vice versa. To prove u n i q u e n e s s we note that if z(t)
is a second w e a k solution,
then t1 / o and h e n c e
dt = 0
~
g e C[O,tl;Z* ]
z(t) = z(t)
Finally, we n o t e that for a s m o o t h e r class of g, say g £ C ~ [O,tl;Z* ] , we have + A*x = g ,
x(t I) = 0
33
and t h e n
{
b) b e c o m e s tI
<~(t)+A*x(t) ,z (t)> at ÷ f
o
which
2.3
Analytic
which
in the
Definition
additional
familiar
definition
important can be
complex
2.27
An a l a l y t i c
plane
Analytic
semigroup
class
continued
for e a c h
of
solution.
continuous as
semigroups
functions
of t i n t o
t > O.
semigroup
T t is
a strongly
continued
continuous
analytically
t e S , and
all
any 0 < e < ~, 3 K > 0
llTtll < ~,
semigroup
with
the
into : ~ e
the
sector
(O,~/2)
t ~ O
z e Z
a n d A T t e ~(Z)
such
IIATtll
that
< ~--~ - Itl'
t ~ S ~-~
2.28
Assume space
a weak
strongly
containing
dtd-- Tt z = A T t z
for
of
analytically
S : larg t I < ~
Theorem
= 0
properties
a) T t can be
c)
dt + < x ( O ) , Z o >
semigroups
A particularly are t h o s e
b)
<x(t),f(t)>
o
is the m o r e
sector
t1
A is a c l o s e d
densely-defined
linear
operator
on a B a n a c h
Z, w i t h
a) R e s o l v e n t some
~
of A, E
(O,½~)
p(A),
is c o n t a i n e d
in t h e
sector
S½~+~
for
a
34
b) ~ M > O independent
of ~, such that
M ' li~(~,A)ll ~ I~-7
C2.23)
then A generates
an analytic
for l
s%~+~
semigroup.
Proof We will define T t via the Dunford (2.24) where
Tt
=
(2~i) -I I eAtR(~,A)dl F
F is a contour
to infinity for ~ e F, olutely
with
in S½~+~_e
and uniformly
and
ATt =
d d-t Tt
The semigroup
and since
small amount
in such a manner ~.
~ eltR (l,A)d~
it follows
=
for T t is proved (2.24) but with
AT t
as follows.
We assume T s is
r shifted to the right by a
F', then
1 { f eI's+ItR(I',A)R(I,A)dl (2~i) 2 'F
dl'
equation
R(~',A)R(~,A)
=
(~-~')-I(R(I' ,A) -R (I ,A) )
we find 2 (2~i)
el' SR (I ' ,A) dl '[elt (l-l') -Id~ ' F
- feXtR(l'A)dl F
But
Hence
feAt(l-~)-idl F TsT t = Tt+ s.
= O
of A, we have
t ~ S _ e.
like
TsT t =
abs-
{ eltlR (l'A)dl
= ~A
TsT t
Targ tl 1 > %z converges
it can be differentiated,
{ eat (I+AR(~,A))d~
'
that
Then the integral
arg I =½~+~-E
In fact using the closedness
to give a new contour
Using the resolvent
from infinity with
= ~1
property
given by a formula
llI+
in S _ E.
d 1 d-~ Tt = ~
Hence
running
arg I = -(½~+~-~)
larg t I < ~
that T t is analytic
integral
and
f eX's[x-X')-Id~) r'
f el'S(l-A')-idl' F'
= -2~ie Is
NOW we change the integration contour by F' = F l t
I.
e l' (t/It I)R((t ,/
Tt = l { Hence from
variable
I to l lt I and denote the new
Then by the Cauchy T h e o r e m
Itl ,A lt1-1
d~,
(2.23)
llTtll ~_ ~M~ AT t =
Also
e x'¢t/Itl) ixl-lldx I <_ K
(2hi) -I f el' (t/Itl)AR((l'/Itl),A)Itl -I dl' F /l'
,I'
= (2~i) -I ] e X'(t/Itl)Ii~iR{I~i
A)-I
) It-1 lax'
F Hence (2.25)
___K
[IATt {I
It[
To prove the strong continuity Ttz - z = ~ 1
~ e It { R ( I , A ) _ I - I } z
= ~1 Hence
Ttz-z ~
and D(A)
Ttz ~ z z ~ D(A)
=
dl
for
TtAZ
as
t ~ O +.
is dense in Z, hence t e S _e
an easy c a l c u l a t i o n ATtZ
d ~
=
yields Ttz
Hence the g e n e r a t o r of T t is e q u a l to A on D(A), to that used in T h e o r e m
for z e D(A),
~ e l t R ( l , A ) A z l -I dl
(2~i) -I f R(~,A)AzX -I dX = O F
But T t is b o u n d e d
For
of Tt, we compute
and a similar
2.10 shows that in fact A generates
argument
T t.
We have seen in the proof of this theorem that if T t is an analytic semigroup with g e n e r a t o r A, then d__zz = Az dt The extra structure condition
'
t > O
'
of analytic
z(t) = Ttz o z(O) = z
for any
z o e Z satisfies
o
semigroups
also enables us to relax the
imposed for a s o l u t i o n of the inhomogeneous problem.
Theorem 2.29 If A generates
an analytic
is H~Ider continuous
s e m i g r o u p T t on a Banach space Z and f(t)
for t > O, with
36
II f(t)-f(s)ll and k < 1.
Then
for any
< L(t-s)k
;
0 < s _< t ,
L > 0
zo e Z
t = Ttz o + f T t _ s f ( s ) d s o
z(t) satisfies (2.26)
z = Az
+ f ,
t > O;
z(O)
= zo
Proof Clearly
w e only n e e d
t = ~ Tt_sf(s)ds o
v(t) satisfies
(2.26).
(2.27) Now
to s h o w
We h a v e
v(t)
t = f Tt_s(f(s)-f(t))ds o
Tt_s(f(s)-f(t)) e D(A) and t llATt-s(f(s)-f[t))llds o
Hence
s i n c e A is closed,
So
(2.28)
For
t,d
e D(A)
t = f ATt_s(f(s)-f(t))ds o
=
(Tt-I)f(t)
with t = I ATt-s(f(s)-f(t))ds o it can be s h o w n t h a t v(t)
Av(t) ( In fact
<
e D(A)
t A ] Tt_sf(t)ds o v (t) E D (A)
(t-sK)L(s_t)kds
2.9 (d) t I Tt-s f(t) ds o
and
t -< ~ o
t I Tt-s(f(s)-f(t))ds o
t Af T t _ s ( f ( s ) - f ( t ) ) d s o But b y T h e o r e m
t + I Tt-sf(t)ds o
+
(Tt-I) f (t)
e D(A)
for f e L 2 ( O , t l ; Z )
> O, we h a v e v[t+~)
and so
lim h~O +
(2.29)
D+v(t)
t+~ = f Tt+~_sf(s)ds o v[t+h)-v(t) h = Av(t)
= T~v(t)
= Av(t)
+ f(t)
+ f(t),
t+~ + f Tt+~_sf(s)ds t so that
)
37 L e m m a 2.30 Av(t)
is c o n t i n u o u s
for
t ~ O
Proof We p r o v e the c o n t i n u i t y on the left for t > O. term in
C l e a r l y the s e c o n d
(2.28) s a t i s f i e s this a s s e r t i o n so we n e e d only c o n s i d e r the
first t e r m for w h i c h we have t-h A ~ Tt_h_s(f(s)-f(t-h))ds o
t - A f Tt_s(f(s)-f(t))ds o
t-h t-h = A [ ~Tt_h_s-Tt_s) (f(s)-f(t-h))ds + A f T t _ s ( f ( t ) - f ( t - h ) ) d s o o t + A f Tt_s(f(s)-f(t))ds t-h Now in a s i m i l a r m a n n e r to the p r o o f of
(2.25)
it is p o s s i b l e to s h o w
that IIA (Tt_Ts)II
-<
K(t-s) ts
'
0
<
s
<
t
Using this and the H ~ i d e r c o n t i n u i t y we o b t a i n the f o l l o w i n g e s t i m a t e for the first t e r m on the right t-h (t_s_h) k KhL ) . . . . as+ O o (t-s) (t-s-h) N o w from T h e o r e m 2.9
as
h +
O+
the s e c o n d term is equal to
Tt(f(t)-f(t-h))
- Th(f(t)-f(t-h))
which can be e s t i m a t e d by L ( M e ~t + l)h k + O
as
h + O+
Finally for the t h i r d term we obtain the e s t i m a t e t f t-h
£ 1 L K (t-s) k ds + O .t.s..
as
h + O+
A s i m i l a r c o n s t r u c t i o n shows that Av(t)
is c o n t i n u o u s
from the right
and so the lemma is proved. Now
(2.29) and lemma 2.30 show that v(t)
left as w e l l and s a t i s f i e s
2.4
Perturbation In a p p l i c a t i o n s
is d i f f e r e n t i a b l e
from the
(2.26).
theory to control problems,
the i n h o m o g e n e o u s
term f in
(2.19) is o f t e n d e t e r m i n e d by a control input and if this is of feedback type, we have f(t) = Bz(t)
38
and we must study t z(t) = Ttz O + ~ Tt_sBZ(s)ds o
(2.30) Theorem
2.31
Let A generate
a strongly
Z, and let B e ~(Z), St, defined
continuous
semigroup
then A+B generates
by
if
continuous
space
semigroup
t Stz ° = Ttz O + ~ Tt_sBSsZodS o
Moreover,
T t on a Banach
a strongly ,
zo e Z
~Tt~ I ~ Me ~t, then
llstlI _< Me(~+mllB11)t Proof First we show that S t defined semigroup.
by
(2.30)
We will seek a solution
(2.31)
St =
~
is a strongly
by the following
continuous
scheme
n St
n=o where (2.32)
Stz o = /tT t _ sBSn-lzods o
It is easy to obtain
the following
n
So the series
St
S t° = T t
,
estimate
by induction
Mn+lllBlln e~t t n
Snt
=
is m a j o r i z e d
by
n=o Me et ~. (Mll~IJt)n n! n=o Thus
S t is absolutely n=o on any compact interval. (2.33)
= Me(~+MIIBll) t
convergent
in the uniform
topology
of ~(Z)
Moreover
StZo =
~ StZ n ° = Stz o ° + n~_1 S z O n=o t = Ttz o + ~ f Tt_sBS~-izodS n=l o
t = Ttz o + ~ Tt-sBSsZo ds So that our construction To prove uniqueness the equations
yields
a solution
we assume
for S t and St gives
of
(2.30)
a second solution
satisfying
So= I.
St' then subtracting
39 t (St-St)Zo = °~ Tt-sB(Ss-Ss)ZodS t (t-s) H(St-St)Zol I < f Me w HBlllI(Ss-Ss)Zoll ds o
Hence Setting
e -mtll(St-St)zol I = g(t), yields t
g~t) < MIIBII I g(s)as o whence
g(t) = 0
from Gronwall's
lemma.
To prove S t is strongly continuous,
we compute for h > 0
t IISt+hZo-Stgoll _< IITt+hZo-TtZoII + f II(Tt+h_s-Tt_s)BSsZoH ds o t+h
÷ f ~l~t÷h_sBSsZot I as t
So u s i n g
the
Lebesque
dominated
convergence
theorem,
the
strong
con-
tinuity
of Tt, and the bounds for T t and St, we see that Stz o is continuous on the right. For the continuity on the left, we have t-h t llSt_hZo-StZoll < llTt_hZo-TtZoll + llf Tt_h_sBSsZo ds - f Tt-sBSsZodSll o o t-h t-h = llTt_hZo-TtZol I + IIf Tt_h_sBSsZo ds - f Tt_h_sBSs+hZodSll o -h t-h
_< ll~t.hZo-Tt~oll ÷ III Tt_h_sBCSs-Ss÷hl~ o dslJ o
o
+
I rlTt_s_hBSs+hZoIIds
-h Using the r i g h t c o n t i n u i t y o f St , t h e s t r o n g c o n t i n u i t y o f T t , and t h e bounds for Tt, S t and the Lebesque dominated convergence theorem we see that S t z o i s
continuous
on t h e
left,
and h e n c e
In order to prove the semigroup property
St
is
strongly
continuous.
st+ s = S t S s, we use
(2.33)
to obtain t+s St+sZo-StSsZ o = Tt+sZ O + f Tt+s_~ABSuZod~ o t t+s
s
s t - f Tt+s_~BSsZod8 - f Tt_~BS~SsZod~ o o o t+s t -- f T t + s - ~ B S ~ Z o d ~ f Tt-~BS~SsZo da s o t = f Tt-~B (Ss+~-SsS ~) Zod~ o
=
/
Tt+s_~BS~Zod~
40 t
Hence
]ISt+sZo-StSsZoll < MI~BII/ e~(t-s)llSs+e-SsS~zJld~ o If
g(t)
= e-~tlISt+sZo-StSsZoll,
g(t)
<_ MIIBII/tg(s)ds 0
So by Gronwall
But
then
s lemma,
g(t)
< g(O)e MIIBIIt
g(O)
= IISsZo-SsZol I = O, so
St+ s = StS s By Theorem 2.9 we know that the infinitesimal generator of S t must be a closed densely defined operator, but although we have motivated the construction
of S t via the operator
A+B is its generator. To see that this t (2.34) f St_e(A+B)Zode = StZo-Z O o ' Now from
(2.29) t
t
A+B, we have yet to prove that is the case we show that z
o
e D(A)
t
f St_~(A÷B)Zod~ = f Tt_~AZod~ ÷ f Tt_~BZod~ o
By Theorem
o o t t +f f Tt_sSSs_ ~ (A+B) ZodSd~ o
2.9 (d) we have t t (/os o) ] St_c~(A+B)Zod0~ = T t Z o - Z o + ~o T t - s B Ss_ e(A+B)zodc~+z d s o
t f St_c~(A+B)Zod~ + z ° i s a s o l u t i o n of (2.30). But we know t h a t o Stz ° is the unique solution o f ( 2 . 3 0 ) s o we h a v e ( 2 , 3 4 ) . Thus
S oZ Hence
lim h+o+
o h
= lim h+o+ =
But A+B is closed
(A+B)z
and densely
S ° (A+B) Zode
(2.35)
perturbation
~ m (O,tl; ~ ( Z ) ) =
z
e D (A) o
o
defined,
In the above theorem we assumed need to consider
,
and so A+B is the generator
that B e ~(Z).
operators { B: B(-)Zo and
However we will
B e ~ (O,tl; ~(Z)), is strongly
measurable
ess sup IIB(t)II~(Z)< ~
0~t ~t I
where }
of S t . also
41
In this case we are e x a m i n i n g a b s t r a c t e v o l u t i o n o p e r a t o r s A+B(t)
of the form
and we w o u l d not expect them to g e n e r a t e o n e - p a r a m e t e r semigroups.
Definition 2.32
Mild evolution operator
Let A(t I) = { (t,s); O ~ s ~ t ~ tl}, then U(t,s):
A(t I) + ~(Z)
is a
mild e v o l u t i o n o p e r a t o r if a)
U(t,t)
= I ,
b)
U(t,r)U(r,s)
c)
U(.,s)
t e [O,t~ = U(t,s)
,
O ~ s ~ t ~ t1
is s t r o n g l y c o n t i n u o u s on
c o n t i n u o u s on A c o n s e q u e n c e of
[S,tl]
and U(t,.)
is s t r o n g l y
[O,t] c) is that
ess sup IIU(t,s)ll
<
~(t l) Clearly if T t is a s t r o n g l y c o n t i n u o u s semigroup,
then Tt_ s is a
mild e v o l u t i o n operator. M o t i v a t e d by
(2.30) we can c o n s t r u c t a m i l d e v o l u t i o n o p e r a t o r w h i c h
is a s s o c i a t e d w i t h the o p e r a t o r A+B(t), by t U(t,s)z o = Tt_sZ O + f Tt_ B ( ~ ) U ( ~ , s ) z o d~ s
(2.36)
The c o n s t r u c t i o n is c a r r i e d out by the f o l l o w i n g scheme (2.37)
Uo(t,s)
= Tt_ s
(2.38)
t U n ( t , s ) z ° = ! Tt_ B ( ~ ) U n _ l ( ~ , s ) z ° d~
and (2.39)
U(t,s)
Using the same m e t h o d
=
~ Un(t,s) n=o
as we used in p r o v i n g T h e o r e m 2.31, we can estab-
lish the f o l l o w i n g Theorem 2.33 If T t is a s t r o n g l y c o n t i n u o u s then
(2.39)
s e m i g r o u p on Z and B E
is the unique s o l u t i o n of
ution o p e r a t o r s
(O,tl; ~ ( Z ) ) ,
(2.36) in the class of m i l d evol-
on Z.
Unfortunately
it is not true that U(t,s)z
so we cannot hope for s o l u t i o n s
is d i f f e r e n t i a b l e
in t and
in the usual sense of the d i f f e r e n t i a l
equation (2.40)
z =
(A+B(t))z
;
z (O) = z
o
e D(A)
Still, we can regard U(t,O)z O as the m i l d s o l u t i o n of isfies the i n t e g r a l e q u a t i o n
(2.41)
t z(t) = Ttz ° + ~o T t - e B ( e ) z ( ~ ) d @
(2.40)
as it sat-
42
Surprisingly Theorem
we can p r o v e
differentiability
in the s e c o n d
variable.
2.34
Under
the a s s u m p t i o n s
of T h e o r e m
2.33,
for
z o ~ D(A),
t (2.42)
f U ( t , o ) ( A + B ( o ) ) z ° do = u ( t , s ) z O - z ° s
(2.43)
~--~ U(t ,s) z o = -U(t,s) (A+B(s))z O
a.e.
Proof Consider
the s y s t e m U'o (t,s) = T t - s t = ! U'n_l(t,o)B(o)To_sZ ° do
U'n(t,S)Zo Then
as in T h e o r e m
2.31,
we can s h o w t h a t
u' (t,s) = ~ Un(t,s) n=o is the u n i q u e
solution
(2.44)
t U' (t,s)z O = T t _ s Z O + f U' ( t , o ) B ( o ) T o _ s Z o d S s
But by i n d u c t i o n ,
U' (t,s) r~
U(t,s) Hence
for
z
o
of
= Un(t,s)
for each n > O, and so
= U' (t,s)
e D (A) , t U(t,~)Az O = Tt_oAz O + f U(t,~)B(~)T~_~AZod8 o
and b o t h
terms
on the r i g h t
are i n t e g r a b l e
t
t
{ U(t,o)AZod~ s
= f Tt_ A Z o d O s
by T h e o r e m
2.9,
so
t t + f f U(t,B)B(~)T8_ s
AZodB
t
=
t T t _ o A Z o d~ + [ I U ( t , 8 ) B ( S ) T s _ o A Z o d ~ s s s
do
I
by F u b i n i ' s
=
d8
theorem
t (Tt_s-I)z o + f U ( t , 8 ) B ( 8 ) ( T ~ _ s - I ) Z o d 8 8 by T h e o r e m
2.9(d)
t So
f U(t,~)(A+B(a))Zode s
= U ( t , S ) Z o - Z O by
(2.44)
and this
implies
(2.431
43
It turns out that this p r o p e r t y is i m p o r t a n t for control and f i l t e r i n g problems so we give o p e r a t o r s w i t h the p r o p e r t y D e f i n i t i o n 2.35
(2.42)
a special name.
Quasi-evolution operators
A q u a s i - e v o l u t i o n o p e r a t o r U(t,s)
is a m i l d e v o l u t i o n o p e r a t o r such that
there exists a n o n - z e r o z for almost all s¢[O,tl]
e Z and a closed linear o p e r a t o r A(s) on Z o satisfying
t U(t,S)Zo-Z o = f U(t,~)A(~)Zod~ s
(2.45)
We denote the set of z o e Z for w h i c h
(2.45) is v a l i d as DA,
and we call
A(t) the quasi g e n e r a t o r of U(t,s). Those quasi e v o l u t i o n o p e r a t o r s w h i c h are also d i f f e r e n t i a b l e first v a r i a b l e are also i m p o r t a n t in a p p l i c a t i o n s D e f i n i t i o n 2.36
in the
and so we define
Almost strong evolution operators
An almost s t r o n g e v o l u t i o n o p e r a t o r is a m i l d e v o l u t i o n o p e r a t o r on Z for w h i c h there exists an a s s o c i a t e d closed linear o p e r a t o r A(t) Z for almost all t e [O,t13 (2.46)
U(t,s)
42.47)
t f A(r)U(r,S)Zodr = s
.for z
o
on
such that
: D(A(s))
+ DCA(t))
for a l m o s t all t>se[O,tl]__
(U(t,s)-I)z o
e D(A(s)) .
(2.46) implies that (2.47)'
~-- U(t,s)z = A ( t ) U ( t , s ) z ° a.e. ~t o
This of course m e a n s that
(2.46) and
(2.47) e v e r y w h e r e
operator in the literature, almost strong concept.
o
e D(A(s))
(2.40) w i l l have a unique s o l u t i o n
z~t~ = U(t,O)z O w h i c h is d i f f e r e n t i a b l e satisfies
for z
almost everywhere.
If U(t,s)
it is c a l l e d a strong e v o l u t i o n
a i t h o u g h for our p u r p o s e s we only use the
We now give s u f f i c i e n t
conditions
for A + B (t) to
generate an a l m o s t s t r o n g e v o l u t i o n operator. Lemma 2.37 Suppose T t is a s t r o n g l y c o n t i n u o u s B ~=
(O,tl; ~ C Z ) )
s e m i g r o u p on Z and
satisfies
Tt_sB(S)
: Z+D(A)
for a l m o s t all t > s e [O,tl]
(2.48) ATt_.B(')
e ~I (O,t;~(Z))
44
that
is A T t _ s B ( s ) z o is s t r o n g l y m e a s u r a b l e
~IIIAT.
B(s)II
~-s
o
ds < ~ ~(Z)
e r a t e d b y A+B(t)
in s ¥ z o e Z and
T h e n the m i l d e v o l u t i o n
operator
U(t,s)
gen-
"
is an a l m o s t
strong evolution
operator.
Proof From
(2.36), U(t,s)
U(~,s)z O = T
Now by
(2.48),
is d e f i n e d
by o
¢ Z
T a _ r B ( r ) U ( r , s ) z O ~ D(A)
and
AT _rB(r)U(r,s)z O
S i n c e A is a c l o s e d
linear
H e n c e we can i n t e g r a t e ,
is i n t e g r a b l e
on
we conclude
that
operator,
A~ T _ r B ( r ) U ( r , S ) Z o d r s
t
z
sZ O + ~ T ~ _ r B ( r ) U ~ r , S ) Z o d r , s
(s,a).
= ~ AT _rB(r)U(r,S)Zodr s
obtaining
e
f Af T _ r B ( r ) U ( r , S ) Z o d r s s
t t d~ = f ~ A T _ r B ( r ) U ( r , s ) z O d~ dr S
r
by F u b i n i ' s
theorem
t = f s
(Tt_r-I)B(r)U(r,S)Zo by T h e o r e m
Hence
from
t = U(t,S)Zo-Tt_sZo - / B(r)U(r,S)Zodr s
A(U(~,s)-T~_s)Zod~ s using
2.9 (d)
(2.36), w e h a v e
t
Thus
dr
Theorem
2.9 (d) again,
t (A+B(~))U(~,S)Zode
= U ( t , s ) z O- z O
s
For other 2.5
examples
Abstract
evolution
In the f o l l o w i n g (2.49) where
of e v o l u t i o n
z =
operators
see C h a p t e r
equations
chapters (A+B(t))z
we shall meet + f(t)
;
equations
It is n a t u r a l
g e n e r a t o r of a s t r o n g l y ~ ( Z ) ) , and f £ L 1 (O,tl;Z).
to d e f i n e
the m i l d
solution
of
t (2.50)
z(t)
of the f o r m
z ( O ) = z o c D(A)
A is the i n f i n i t e s i m a l
g r o u p Tt, B e ~ ( O , t l ;
9.
= U(t,O)z o + f U(t,s)f(s)ds o
continuous
(2.49)
to be
semi-
45 where U(t,s)is
the quasi evolution
Then the following perty
(2.43)
result
operator g e n e r a t e d by
is proved e x a c t l y
for almost strong e v o l u t i o n
A+B(t).
as Lemma 2.22 using pro-
operators.
Theorem 2.38 If U(t,s)
is an almost strong e v o l u t i o n
for almost all t > s c [O,tl~ then
(2.50)
operator
and AU(t,.)f(.)
is the unique solution
of
and U(t,s)f(s)e
D(A)
is Bochner integrable,
(2.49).
From this we can also deduce the useful corollary Corollary
2.39
If T t and B(t) the unique
satisfy the assumptions
solution of
(2.49) p r o v i d e d
for almost all t > s £ [O,tl] , and
of Lemma 2.37,
then
that Ttz ° and Tt_sf(S)
ATt_sf(S)
is Bochner
(2.50)
is
e D(A)
integrable
on
(O,t). Proof Now from the proof of Lemma 2.37, AU(t,s)f(s) and this is m e a s u r a b l e
= ATt_sf(S)
t
Z ÷ D(A)
and
t + f ATt_~B(~)U(~,s)f(s)d~ s
in s, and
t ~ llATt_sf(S)llds < ~ o
so
U(t,s):
by assumption.
t
t t
f Ill ATt_~B(~)U(~,s)f(s)d~llds <- f f 11ATt_J(~)lllI~(~,s)lillf(s)I1d~ ds O
S
O S t t <_ const ~ IIATt_ B(~)~d~f IIf(s)llds o O < ~ by
(2.48)
and D e f i n i t i o n
2.32(c).
t Hence
f llAU(t,s)f(s)llds < ~ and the assumptions o
of T h e o r e m 2.38 are
satisfied. 2.6
Examples
Example
2.40
A large class of partial type)
differential
can be associated w i t h analytic
consider
a special
case of Example
Hilbert space H and R(~o,A) that R(~,A)
is compact
equations
semigroups.
(of the p a r a b o l i c In this example we
2.4 w h e r e A is self adjoint on a
is compact
for all I e p(A)
for some 10 e p(A). (see [i0])
and also
This implies
46 a)
B an infinite
sequence
{lj} of distinct e i g e n v a l u e s
as j + ~, each with finite m u l t i p l i c i t y of the c o r r e s p o n d i n g A generates
eigenmanifold.
a strongly
and can be ordered
continuou~
. • . I n <...<
llj[ ÷ ~
Moreover
semigroup,
since we are also assuming the lj are b o u n d e d above
12 < 1 1 < const.
b) 3 a complete o r t h o n o r m a l c) the s p e c t r u m consists
of A,
rj equal to the d i m e n s i o n a l i t y
set {¢9 k} of e i g e n v e c t o r s
of A.
only of the point spectrum
d) from the unique e x p a n s i o n r. z =
~ ~3<s,,jk>,j k j=l k=l
one gets for z e D(A)
rj Az--
j=l
k=l
e) for k not in Pq(A) R(I,A) z = f) the s e m i g r o u p
CJk>¢jk
and each z ¢ H r. 3
1
~
l---~j ~
j=l
k=l
~k ~
g e n e r a t e d by A is given by X t e 3
Ttz = j=l
r. ] ~ @j k , k:l
These results may be found in [5] or
z e H, t _> O
[iO]. M o r e o v e r
it is easy to obtain
the estimate HR(I,A)I I ~ M SO the semigroup T t is analytic. As a special Example
1.2.
case we c o n s i d e r the p a r t i a l d i f f e r e n t i a l
k~2z_~ AZ = ~ l ~ X 2 with
e q u a t i o n of
If we set
D(A)
=
z e L2(O,I):
Then A is self adjoint with PCll j = ~ +
~jl
z PCl ~ £ L2(O,I) , ~ ~x 2
compact resolvent. (J-l)2z2k
1 = - cos(J-l)~x /2
The eigenvalues
and e i g e n v e c t o r s j > i,
= 0 at x = 0,i
#iI = 1
are
47 Example
2.41
As a s p e c i a l p(~)
= a(~)
case of E x a m p l e
= 1
¥ ~ e [O,i],
(2.51)
we c o n s i d e r
Example
1.5 w i t h
then
ztt = z ~ z(O,t)
If H = L 2 [ O , 1 ] ,
and Az = - z ~ , D(A)
then A
2.16,
= z(1,t)
= O
z e D(A), = H 2 (O,i)
where
n H i (0,I) o
= A, and 1
1
H o So we may
apply
the r e s u l t s
of E x a m p l e
2.16
to c o n c l u d e
that
the o p e r -
ator
D(~)
on ~ =
D(A½)× H g e n e r a t e s
inner p r o d u c t
on ~ i s
<w,~>
(2.52)
Tt
continuous
e Hl(O,l)x o
1 = f w1~(~)wi~(~)d~ o
L2(O,I)
semigroup
Tt .
equivalent
Here
the
to
1 + f w2(~)~2(~)d ~ o
i 2 [ < w l,~n > cos n~t + 1 <w 2'#n > sin n n t ] ~ n
= 2 [ - n z < w l , ~ n > sin n~t + <w2,~n>
w2 where
x D ( A ½)
a strongly
for w , w
[wl]
and
= D(A)
cos n ~ t ] ~ n
~n = sin n ~ .
The m i l d
solution
of
(2.51)
is
= Tt z
for z O e H oz (O,i) , Zot e L2[O,I] . then this
solution
Zot
If i n s t e a d
w i l l be a s t r o n g
solution.
Zo e D(A),
Zot e Hl[O,l],o
48
Example
2.42
Delay equations
We consider
the linear system on [O,tl] n
(2.53)
x(t)
= A x(t) o
+
Z A. i=l i
i
x(t+@ i) ; t+8 i > 0 h(t+@i ) ; t+@i < O
o (x(t+8) + ~b A°l(8) I h(t+8)
; t+8 >_ O d8
; t+% < 0
x (0) = h (0) where -b ~ 8 n < 8n_ 1 < ... < 81 < O and b is a positive Ao,Ai;
i=l,2,...,n
e ~(H), where H is a Hilbert
space,
number. and
Aol e C(-b,O; ~(H)). It is p o s s i b l e to define this system on C(-b,O;H), trol a p p l i c a t i o n s bert space.
functions
on
be the space of measurable,
[-b,O],
then ~ 2 ( - b , O ; H )
in ~2(-b,O;H)
however
for con-
useful to define the system on a Hil-
So we introduce the s p a c e ~ 2 ( - b , O ; H )
Let ~ ( - b , O ; H ) of functions
it is sometimes
as follows:
square
integrable
is the space of e q u i v a l e n c e
under the e q u i v a l e n c e
H valued classes
relation
o j~2 = H + [ dt -b H and is i s o m e t r i c a l l y Detailed authors
isomorphic
studies of Equation
to the space H x L2(-b,O;H). (2.53) have been made by a n u m b e r of
[13, [2], [43 and we summarize
introduce
the space AC2(-~O;H)
functions
whose d e r i v a t i v e s
their main conclusions.
as the space of absolutely
are square
integrable,
continuous
with the inner pro-
duct AC2 AC 2 is a Hilbert
x(t) N O W we introduce
= f°< df d-9 at ~b dr" dt>H
space,
and in
= ~(t,h)
~]
it has been shown that
e AC2(O,tl;H)
(h ° x) t e ~ 2 (-b,O;H)
for h e v~2(-b,O;H)
d e f i n e d by
I x(t+@) ; t+@ > 0 (h or)t(8)
= h(t+8) ; t+8
< O
First we
49
for all 8 s [-b,O]
and set
(2.55) where
(Tth) (8) = x is the s o l u t i o n
of
(h o x)t(8)
(2.53)
satisfying
2.54.
Then
a) T t e i(J~2 (-b,O;H)) b) T O = I in J~2 c) T t is s t r o n g l y
continuous
d) T t : A C 2 ( - b , O ; H ) ~
AC2(-b,O;H)
e) for t > b T t : 6 2 ÷ AC 2 f) T t + s = TtT s If we set group,
z(t)
we k n o w
= Tth,
that
then
since T t is a s t r o n g l y
z satisfies = Az
for h e D(A).
From
(2.53)
an a b s t r a c t ; z(O)
(Ah) (8) =
and D(A)
continuous equation
= h
it is c l e a r
that
n
I A o h (O) +
evolution
o
7. Aih(0i) i=l
+ ]_ b A o l ( f ) h ( 8 ) d 8
dh dO
; 8 = 0
;e#o
= H I [-b,O;H].
L a t e r we shall
(2.56)
consider
z(t)
the i n h o m o g e n e o u s
equation
t = Tth + f T t _ s ~ ( S ) d s O
where ~ is d e g e n e r a t e
in the s e n s e
that
(~(s))(0) =I f(s) ; 0 = o o ;0~o f e L2[O,T;H],
then
[4] z(t)(0) n
(t)
= AoX(t)
+
is the s o l u t i o n
I x(t+%i)
Z A. ~ i=l z h(t+0i ) o
x(o)
= h(O)
> O
i t+8 i
< O
(x(t+8)
+ f A°l(e) I -b
t+8.
h (t+8)
t+0
d0 ;
t+0
of
> O
-
< O
+ f(t)
semion ~ 2
50 SO the solution of inhomogeneous delay equations may be cast into the abstract form (2.21).
References ~i] Banks, H.T.
The representation of solutions of linear functional differential equations,
J. Diff. Eqns., 5 (1969)
pp.399-410. [2] Borisovic, J.G. and Turbabin, A.S.
On the Cauchy problem for linear
non-homogeneous differential equations with retarded arguments, [3] Butzer, P.L. and Berens, H. ions,
Soviet Math, Dokl., iO (1969), pp.401-405. Semigroups of operators and approximat-
Springer Verlag, 1967.
[4] Delfour, M.C. and Mitter, S.K.
Hereditary dlfferential systems with
constant delays I; General case, 12 (1972), pp.213-235. [5] Dunford, N. and Schwartz, J.T.
Linear operators,
J. Diff. Eqns., Interscience
1959, 1963. [6] Hille, E. and Philllps, R.S.
Functional analysls and semigroups,
Amer. Math. Soc. Coil. Publ., [7] Kato, T.
31 (1957).
Perturbation theory of linear operators,
Springer Verlag,
1966. [8] Kisynski, J.
Applications of semigroups to partial differential equations,
Course on control theory and topics in
functional analysis, [9] Ladas, G.E. and Lashmikantham, V.
Trieste, 1974.
Differential equations in abstract
spaces, Academic Press, 1972. [10]Taylor, A.E. Introduction to functional analysis, John Wiley, 1958. [ll]Yoslda, K. Functional analysls, Springer Verlag, 1966.
CHAPTER 3
CONTROLLABILITY,
OBSERVABILITYAND
STABILITY
In this chapter we c o n s i d e r the c o n t r o l l e d s y s t e m
(3.1)
~ = Az + Bu, z(O) = z
o
w h e r e A g e n e r a t e s a s t r o n g l y c o n t i n u o u s s e m i g r o u p T t on a B a n a c h space Z, and B is a b o u n d e d o p e r a t o r
f r o m a c o n t r o l B a n a c h space U to Z.
H o w e v e r if z is a strict s o l u t i o n of t ~ [O,tl]
(3.1) then z(t)
c D(A)
so in the general c a s e w h e r e A is u n b o u n d e d ,
means that the s y s t e m cannot be s t e e r e d to all of Z.
for all
D(A)
~ Z, w h i c h
We will,
therefore
choose to w o r k w i t h the m i l d solution. (Definition 2.23). t (3.2)
z(t)
= Ttz O + f
Tt_sBU(s)ds o
and i n t r o d u c e the f o l l o w i n g c o n c e p t of exact c o n t r o l l a b i l i t y .
Definition
3.1
E x a c t c o n t r o l l a b i l i t y on
Given any two points Zo,Z 1 e Z we say on
[O,tl]
[O,t~ (3.2) is e x a c t l y c o n t r o l l a b l e
if there exists a c o n t r o l u e L P [ O , t l ; U ] such that z(tl)
= zI
First we prove a n e g a t i v e r e s u l t Proposition
3.2
L e t Z,U be B a n a c h spaces and B e in the u n i f o r m t o p o l o g y of o p e r a t o r s sional subspaces of
~(U,Z).
TsBU(s)ds o
f r o m L ! [O,tl;U~
be such that B is a limit
{B n} w h i c h b e l o n g to finite dimen-
T h e n the o p e r a t o r
t1 Au = ;
~(U,Z)
into Z is compact.
52 Proof In order to prove
this p r o p o s i t i o n we first assume B is of the form Bu = bu
w h e r e b ~ Z and u
e U .
For some integer N we set ti
itl -
and define
(u)
N
i
O,...,N
an o p e r a t o r FN: L * [ O , t l ; U 3 ~ Z by N
ti
FNU =
Ttib i=l
C l e a r l y F N is compact logy t o
=
an
operator
f
w
u
(u(s))ds
ti-I
and we show that F N converges
F
as
N ~ ~,
in the u n i f o r m topo-
where
t 1
F U = f
TsbU
(U(s))ds
O We have ti ;ti-li]Tsb-Tt ibllZ ]lu*ilflu (s)ilU ds
IfFNu-F®uflz _< z
i=l
But T t is a strongly and N s u f f i c i e n t l y
continuous
semigroup
on Z so that for s E(ti-l,t i)
large
t[Tsb - Ttibllz
ti
iIFNu -
.N ! F~Ullz <_ ~IIu lli~ 1 i-111 sOs>flu ds ~flu*tliluilL~E0,tl;U]
So F hence F
is the u n i f o r m is compact.
dimensional
limit of a sequence
of compact operators
We may easily extend this argument
o p e r a t o r Bn, so the operator
and
to any finite
A n defined by
t1 hnU = S
TsBnU(S)ds
O
is compact.
We now show that A n ÷ A in the u n i f o r m t o p o l o g y as n + ~.
53
We h a v e t1 llAU-AnUIIz <_ f IITs(B-Bn)U(S)IIzdS O t1 <
sup
HTsl I IIB-Bnl I f llu(s)ll ds
- s~ [O,tl] But B
n
~ B in the u n i f o r m topology,
o
thus A
n
÷ A in the u n i f o r m t o p o l o g y
and so A is compact. We note A can be e x p r e s s e d by t1 Au = ~ T t l _ s B U ( t l - S ) d s
Hence if B is the u n i f o r m limit of a s e q u e n c e of finite d i m e n s i o n a l operators basis),
(an e x a m p l e is a compact o p e r a t o r on a space w i t h a Schauder
(3.2) cannot be exactly controllable.
n a t u r a l e x t e n s i o n of the finite d i m e n s i o n a l
This indicates
that the
concept of c o n t r o l l a b i l i t y
to i n f i n i t e d i m e n s i o n s may be too s t r o n g for m a n y i n f i n i t e d i m e n s i o n a l systems,
even w h e n we consider
the m i l d s o l u t i o n
(3.2).
we w i l l i n t r o d u c e a n u m b e r of d i f f e r e n t d e f i n i t i o n s
In this chapter
of c o n t r o l l a b i l i t y ,
but b e f o r e we do this we prove some t h e o r e m s w h i c h w i l l enable us to o b t a i n n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s
for c o n t r o l l a b i l i t y and link
t h e m n a t u r a l l y w i t h the c o r r e s p o n d i n g c o n c e p t s of o b s e r v a b i l i t y . T h e o r e m 3.3 Let F e ~(V,Z),
G e ~(W,Z), Range (F)
w h e r e V,W,Z are B a n a c h spaces, C
then if
Range (G)
there exists Y > O, such that
ll~'z*II. V
_< ~JIG*z*ll. W
Proof First we assume that G is a l-1 mapping, defined.
then G-I:
R(G) + W is w e l l
M o r e o v e ~ G-IF is a c l o s e d t r a n s f o r m a t i o n from V into W and is
t h e r e f o r e bounded.
Thus the adjoint of G-IF is bounded,
exists y > O, such that
and there
54
'"ll(G-l~)*w*ll.
(3.3)
<_
"fllw*ll" '
V
.
w w
~ w
W
*
Now
for
any
z
E Z , we
let w
= G z , then
< --(G-IF)*w*,v>
= <(G-IF)*G*z*,v>
.
V
for all v e V , V
,V =
<
G
*Z
*, G-1Fv
>
,
W =
*
;
GG-1Fv>
,V
,W
,
Z ,Z =
,Fv>
.
Z ,Z =
, V
Thus
(3.3)
implies
ll~*JII. For [W]
^
the
general
e W being
the
<_ YllG*z*ll.
V
W
case w e
define
equivalence
II[w]ll^ then
define
the
first
of w + ~
G (~) = O
the
part
C
it
follows
Theorem
definitions
theorem
that
W
of W a n d G it is e a s y
=
to s h o w
IIG*~*II . W
is p r o v e d .
3.4
If F e ~ ( Z , V ) , exists
and
_< Y ll~*z*ll^.
W
so the
O,
R a n g e (G)
of the p r o o f
ll~*z*II^. and
G(~)=
W
V from
W = W / k e r (G) w i t h
~Ew] = Gw, a n d so G is a i-i m a p on W, ^ with
ll~*z*ll , But
space
for w h i c h
inf IIw+'~ll
=
R a n g e (F) From
the B a n a c h
class
w
We
,V
~ > O,
G e ~(Z,W), such
where
V,W,Z
are
that V
W
then *
Range(F
*
) C
Range(G
)
Banach
spaces,
and there
55
Proof *
Assume
*
z
*
= F v
equivalently
we
*
, then are
(3.5)
our
looking
,Fz>
for w .
V The
identity
(3.5)
such
= <W
defines
,Gz>
a linear
is a s e q u e n c e
such
that
*
z
*
= G w
, or
V
z e Z
,W
functional
,
where
V i f { z n}
, W
,Fz>
such
that
,V
fw* (y) = < v
Moreover
*
a i m is to f i n d w
on Range(G)
by
y = Gz
,v Gz n ÷ O a s n ~ ~, t h e n b y
that
(3.4)
*
Fz n + 0 as n ~ ~ Thus
w
and so <w
is a c o n t i n u o u s
Hahn-Banach is w e l l
defined
Corollary If V , W , Z Theorem
theorem
,GZn>w,,w
linear
= v*,v
functional
fw* c a n b e e x t e n d e d
by
o n R a n g e (G) a n d
to the whole
so by
of W.
the
Hence w
3.5 are reflexive
3.3
Banach
and Theorem
spaces
3.4 y i e l d
(a)
R a n g e (F)
(b)
there
and F £ ~(V,Z),
the C
equivalence
G e ~(W,Z)
of the following
R a n g e (G)
exists
7 > O,
ll*z*llV .
Let
as n ÷
(3.5).
(3.6)
Theorem
÷ 0
such
that
IIG* *II , W
3.6
F e ~(V,Z),
following
G e ~(W,Z),
conditions
where
V,W,Z
are Banach
spaces,
then
the
are equivalent (a)
ker(G
) C
(b)
Range(G)
ker(F
)
(3.7) D Range(F)
Proof *
Assume there
ker(G must
construct
*
) C ker(F
exist
), b u t R a n g e ( G )
an e l e m e n t
a functional
on Range(G).
z
does
z e Range(F),
~ 0 on
Z
such
not
but that
Thus
contain
Range(F).
z ~ Range(G),
, Z
~ O and ,Z
.
,Gw>
. Z
and hence
z
e ker(G
).
Now
= O
¥ w
e W
,Z if z
e ker(F
), w e w o u l d
so w e
have
Then can z* = O
58
,Fv>
Since
z c Range
z -- l i m FV n . n+~ For
the
(F), t h e r e ,
SO < z
,z>
=0
.
Z
exists
,
= O,
we
suppose
,Gv>
a sequence contrary
{v n } i n
to
our
= O on Range
(G),
(3.7)(b)
that
,
V such
that
assumption.
= O
holds
and
G z
= O.
Then
¥ v e V.
,Z
Z z
¢ V.
Z tZ
converse,
Hence
Wv
,Z
a n d so a l s o o n R a n g e
(F). T h u s
,
= O; t h a t
In t h e
sequel
so the major We will
,Fv>
is z the
tools
= 0
e ker
spaces
in t h e
be particularly
¥ v e V
(F).
V,W,Z
will
analysis
interested
be reflexive
will
Banach
be Corollary
in t h e m a p
spaces
and
3.5 a n d T h e o r e m
G defined
3.6.
by
t1 G u = of
where
u e L P [ O , t l ; U ] 1 < p < ~, T t is a s t r o n g l y
a reflexive B e
Banach
~(U,Z).
Then
space
and we
can use
exact
Theorem
continuous Banach
semigroup
space,
3.5
to o b t a i n
the
following
equivalent
condition
3.7
is e x a c t l y
1 < p < ~ and U,Z controllable
are
iff there
reflexive exists
Banach
y > O,
¥11B*T~.)z*llLq[O,tl;U* ] t llz*ilZ. ~1 + ~1 = i.
on
and
~(LP[O,tl;U],Z)
Corollary
(3.8)
where
a reflexive
controllability.
If u E L P LjO ,-t l~; U (3.2)
Z, U is a l s o
clearly G e
for
Ttl-sBU(s)ds
spaces,
such
that
then
57
Proof We apply C o r o l l a r y
3.5 w i t h F = I, the i d e n t i t y on Z, W = L P [ O , t l ; U ] ,
Then W
and V = Z.
= L q [ O , t l ; U * ] a n d Tt
group on Z (see T h e o r e m 2.20).
is
a strongly
continuous
semi-
We n e e d to c a l c u l a t e G , and for this we
have W
t 1
*
, Z
=
Ttl_sBU(s)ds>
o
tl =
f
, ,
,
Ttl_sz
o
= B Ttl_sz
Thus G z
ence
of
. Z ,Z
, and applying
,u(s)> , ds U ,U
Corollary
3 . 5 we o b t a i n
Range(G)
O
Z
But TtlZ O is a fixed point in Z for any given Zo, and so v a l e n t to exact c o n t r o l l a b i l i t y on
(3.9) is equi-
[O,t~.
3.8
We c o n s i d e r the c o n t r o l l e d w a v e e q u a t i o n of E x a m p l e ~(~)
equival-
(3.8) and
(3.9)
Example
the
1.5 w i t h p(~) =
= i, then ztt = z ~
(3.10)
z(O,t)
+ u
= z(l,t)
: O
The a n a l y s i s of the u n c o n t r o l l e d s y s t e m has b e e n c a r r i e d out in E x a m p l e 2.41, so that we may a b s t r a c t w =~w
where B = given by
I u, and ~
(3.10) to obtain +Bu
generates a strongly continuous semigroup T t
(2.52) [Z2[<wl,~n
>
i < Hcos n ~ t + ~ w 2 , ~ n > H S i n n~t]~ n
LE2[-n~<wl,%n>HSin
on a H i l b e r t space ~ ,
n~t + < w 2 , % n > H C O S n~t]~ n
w i t h inner p r o d u c t
]
J
58 1 <w,w>~ = ~o wl~(~)wl~(~)d~ +
]'
1 w 2(~)w2(~)d~
O and ~n = sin n ~ ,
H = L2[O,I].
It is easy to show that T t = T_t , and B = [O,~, so that if u e L2[O,tl;H ~ the system will be exactly controllable iff there exists 7 > O, such that
~ll~*T~.)WllL2[O,tx;~ 3 a llwl[~
(3.11)
Calculation of the terms in (3.11) yields I 72
(t ? n2~ 2 <Wl,#n> ~
+
sin 2n~tl~ 2n~ / + <Wl'~n>H<W2'$n>H (l-c°s 2nntl)
1
<wz,
sin 2nntl~ 1 n
~\
±
+
~
,
/
-
llwlj
But {IwIl~is equivalent to {Z n ~ 2 < W l , ~ n > ~ + <w2,$n>~} ½ and so T in (3.11) can be found if 4
~t/i- sin2 2n~t1\ 4qZ-"~ ~-)nZ~2 > (1-cos 2n~tl)2
and
n = 1,2,...
I
t I > I sin 2nzt 1
But this reduces to t I >
I sin n~t 1 nw
and so the system is exactly controll-
able on [O,tl] for any t I > O. Example
3.9
Consider the controlled diffusion equation Zt = Z ~
+ U
z(O,t) = z(l,t) = 0
59
The semigroup associated with the system is given by 1
Ttz = Z2e-n2~2tsin n ~
on L2[O,IJ. condition
If u £ L2[O,tI,L2[O,I]~
for
(3.12)
exact
controllability
l
-~ ;:2 k(1-e-2n~tlh2n-q= q /
But 11~ I1~,~[O,lJ
[ sin n~y z(y)dy o
then B = I, and B = I, so that the is
the
existence
Xsin n~y ~(y)dy
of
y > O,
such
that
-> Ilzll~[O,X].
is equivalent to 1 [E2(f sin n~y z(y)dy)2]% O
and no such y exists which satisfies
(3.12), and hence the system is
never exactly controllable on [O,tl] for any t I.
We may however ask
whether or not the system is exactly controllable to some subspace of L2[O,I]. For example, if in Corollary 3.5 we set F* to be the injection *
of Z into V
w
where V
is normed by
Ilv*(IV2* = [ ~
then
7~
1 (~ sin sty v*(y)dy) 2
(3.12) is replaced by
_~z~±-e2nz~ 2
±)|Sk ~ sln" n~y z(y)d
_> ~ n ~ 2
sin nny z(y)dy
But the left hand side is greater than
and so the system is exactly controllable to a subspace V of L 2 [O,i] in any time t I > O, where ffvf[~ : ~ n2~ 2
i.e. V = HI(o,1).
n~y v(y)dy
60 The above example i n d i c a t e s that in those cases w h e r e it is not possible to e x a c t l y control the s y s t e m to all points in Z, it may be p o s s i b l e to control the s y s t e m to points w h i c h form a dense set in Z.
So we
i n t r o d u c e the f o l l o w i n g
Definition
3.10
We say that
Approximate controllability (3.2) is a p p r o x i m a t e l y c o n t r o l l a b l e on Range(G)
Thus if
[O,tl]
if
= Z.
(3.2) is a p p r o x i m a t e l y c o n t r o l l a b l e
for any z I e Z, and any
s > O, there exists a control u e L P [ O , t l ; U ] such that
llz(tl) - ZllIz_< ~ T h e o r e m 3.11 (3.2) is a p p r o x i m a t e l y c o n t r o l l a b l e
(3.13)
B Ttz
implies z
= O
on
[O,tl]
iff
on O _< t _< t I
= O.
Proof We a p p l y T h e o r e m
3.6
w i t h F = I, W = L P LJO ,'t l-; U
1 < p < ~, V = Z,
then since Range (F) = Z we see that
(3.2) is a p p r o x i m a t e l y c o n t r o l l a b l e iff ker(G
Computing G
) C ker(I
)
as in T h e o r e m 3.7, we see that the above is e q u i v a l e n t to
(3.13), w h e r e the e q u a l i t y is taken in L q [ O , t l ; U 3 . But
Z is
(3.13)
reflexive
may be
and
interpreted
so
Tt
is
in
U .
a
strongly
continuous
semigroup
and
hence
E x a m p l e 3.12 We c o n s i d e r a special case of the s e m i g r o u p c o n s i d e r e d in E x a m p l e 2.12, where
Z = H a real s e p a r a b l e H i l b e r t space and A has e i g e n f u n c t i o n s ~nk
61 corresponding
to e i g e n v a l u e s
In(ll
> 12...) r
Ttz =
We a s s u m e
the c o n t r o l
of m u l t i p l i c i t y
Then
n
Z elnt Z H n=l k=l
system
r n.
~nk
is of the form m
(3.14)
~ = Az +
Z b u p=l P p
w h e r e b l , . . . , b m e H, and Up e LP[O,tlJ~m = p=l Z bpUp,
B {mU l), ... .., u
u =
1 < p < ~.
Thus
if w e set
(Ul,. .. ,Um) C U = R m, then u ~ n P [ O , t l l U ]
Moreover B z =
Hence
the m i l d
iff
of
(3.14)
is a p p r o x i m a t e l y
r n Z e xnt Z < b p , ~ n k > H < Z * ,~nk > = O n=l k=l
(3.15)
implies Using
solution
(_ < b l , Z > H , . . . < b m , Z > H )
z
The m i l d
p = l...m
w e are a b l e
[O,tl]
t e ___[O,t17
to p r o v e
3.13 solution
of
(3.14)
is a p p r o x i m a t e l y
controllable
iff rank Bj = rj where
on
= O.
(3.15)
Proposition
controllable
Bj is the m a t r i x
H B
3
:
:
•
H
... < b m , ~ j l > H •
H H --- < b m , ~ j r j > H 33 33
Proof We n e e d the f o l l o w i n g
lemma.
on
[O,tl]
62
Lemma
3.14
(3.16)
1.t e 3 epj
Z j=l
= 0
; t e
[O,tl],
P = 1
... m
iff epj
= O
; p = l,...,m,
j = i,...,=.
Proof Because assume
of t h e
analyticity
1 1 < O,
we may
(3.16)
for
all
t ~ O.
lit
write
~PJ
I<-
lj = 1 1 + ~j,
e11t[
pl +
Hence
j=l
with
-
PJ
Gj
I
< O,
o.t j7
~ j=2
e 3 ~p
and
(3.16)
=0
as
t >O
~jt ~pl
and
as t ~ ~, w e
find
The
general
can
Assume
case ...
+
~ e j =2
~pl be
= O ~PJ
= O.
In a s i m i l a r
reduced
1 2 ~ 1 1 ~ c,
and
to t h e set
case
manner where
[9 = l j - c ,
then
we
can
show
I 1 < O as (3.16)
~ j=l
or
e
= O
becomes
t > O
-
~PJ
ljt Z j=l
e
= O
t > O
~PJ
as r e q u i r e d . t h e Lermaa
(3.14) r
z
approximately
controllable
iff
n
Z k=l implies
is
= O.
H
That
is
,¢nk>H
iff
= O
aij
follows:
~jt e ct
By
First
then
j=l and we may
extend
p = 1 ... m r n = 1 ...
= O.
63
B~v. 33 implies
So if
z
= O, w h e r e
(3.14)
vj =
= O,
j = 112,...
[ H ...
is n o t a p p r o x i m a t e l y
,$ r.>H] 3 3
controllable,
there exists
a vj ~ O,
such that B~vj = O and so r a n k Bj ~ rj. Conversely vj =
if r a n k Bj ~ rj for s o m e J, t h e n t h e r e e x i s t s
[Vjl,...,VjrjS'
~ O
such that B jI v j
So w e c a n c o n s t r u c t
Then
r
a non-zero
z
=
O
e H s u c h that
,$jk>H = O
,~Jk>H = Vjk
n
if j ~ J
k = 1 ... rj
, n ~ J,
H H = 0
k=l rj
< b p , $ j k > H
,~Jk>H =
V j k
So if r a n k Bj ~ rj
(3.14)
rank Bj ~ rj is e q u i v a l e n t and this proves
O,
(3.14) b e i n g
not
controllable.
approximately
Thus
controllable,
the proposition.
W e n o t e t h a t the p r o p o s i t i o n is at l e a s t
As a s p e c i a l
=
is n o t a p p r o x i m a t e l y
to
p = 1 ... m
k=l
= B j! V j
Laplacian,
p = 1 ... m
rj
k=l
required
a
implies
t h a t the n u m b e r
that of the h i g h e s t
c a s e of this e x a m p l e w e
multiplicity
of c o n t r o l s
of the e i g e n v a l u e s .
let H = L 2 [ O , I ] ,
Az = ~z,
the
and D(A)
= {z c H, Az
T h e n A is s e l f a d j o i n t elgenvectors
with
are In = -n2~2,and
£ H, z = O at x = O,i}
compact
resolvent
~jl = s i n jzx,
and the e i g e n v a l u e s
rj = i.
So the m i l d
and sol-
64
ution w i l l be a p p r o x i m a t e l y c o n t r o l l a b l e w i t h a single control if, for example 1 b l ( ~ ) s i n nz~ d~ ~ O
¥ n = 1
o
For finite d i m e n s i o n a l systems the c o n c e p t s of a p p r o x i m a t e and exact c o n t r o l l a b i l i t y are equivalent,
and can be c h e c k e d via a r a n k c o n d i t i o n
i n v o l v i n g the o p e r a t o r s A , B ( s e e D e f i n i t i o n
1.3) o
For m o s t systems the
o p e r a t o r s A and B will be given r a t h e r than the s e m i g r o u p Tt, so it is i m p o r t a n t to attempt to o b t a i n c o n d i t i o n s the o p e r a t o r s A and B w h i c h g e n e r a l i s e c e e d in this d i r e c t i o n in two ways;
for c o n t r o l l a b i l i t y
the r a n k condition.
involving
We can
pro-
e i t h e r by d i f f e r e n t i a t i o n of the
s e m i g r o u p w h i c h w i l l lead to domain problems,
or use the formula
(see
L e m m a 2.11). R(~,A)z = f e-~tTt z at o
T h e o r e m 3.15 The s y s t e m
(3.2) is a p p r o x i m a t e l y c o n t r o l l a b l e
in f i n i t e time iff the
system (3.17)
z (t) = e
R (lo,A)t
Zo+
]teR (lo,A) (t-s) Bu (s)d s o
is a p p r o x i m a t e l y c o n t r o l l a b l e vent of A and I
O
in finite time.
Here R(lo,A)
is the resol-
is any p o i n t in the p a r t of r e s o l v e n t set w h i c h contains
Re I > e, w h e r e ~ is d e f i n e d by
(2.14).
By finite time c o n t r o l l a b i l i t y
we m e a n there exists a time t I for each Zo,Z 1 and it is not f i x e d apriori. Proof T h e o r e m 3.11 can e a s i l y be e x t e n d e d to a l l o w for the finite time assumption,
and we h a v e to p r o v e the e q u i v a l e n c e of the s t a t e m e n t s
(3.18)
B Ttz
= 0
t > O implies z
= O
and (3.19)
Moreover,
B*e R
since R
(~o'A)tz * = O
t _> O implies z*
(lo,A) is a b o u n d e d o p e r a t o r
=
O
65
.
eR
tn ~-., [R*(Io,A)] n
~
(lo,A)t =
7 n=o
and
(3.19)
is e q u i v a l e n t
(3.20)
First
to
B*[R*(Io,A)J~~
suppose
B Ttz
= O
[R*(Io'A)]nz*
(see the H i l l e - Y o s i d a
n * z = O
n = O,1...
t ~ O for s o m e
= Rn(lo'A*)z
Theorem
2.10)
*
=
contradicts
Now assume
~ O.
1 (n-l;:
it f o l l o w s
B *JR* (Io,A) ] nz* = 0 and this
z
implies
Z
*
O
Then since
I o
~tn-le-'lotT~
z*dt n=l...
that
n = O,1,2,...
(3.20).
t h a t B * R n ( X O , A * )z * = O
n = 0,1,2,...
for some
z , ~ O.
Since R ( I , A , )z , =
(i_io) n R n + l ( l o , A *) z ,
~ n=o
we have
first for suitably
small
I, a n d t h e n by a n a l y t i c
for all I w i t h Re ~ > B R(I,A
)z
= 0
But u* U = ; e - X t < B * T t z* ,U>u* u dt
o for all
u e U, z
e Z , if Re I > ~.
~ e o This
implies
-lt
(see D u n f o r d
*
*
*
,U>u* U d t
p.626)
,U>u* U = O
t > O
B Ttz contradicts
the
= 0
and S c h w a r t z
or
which
Hence
assumption.
= O
Re I > ~
continuation
66
A sufficient rank
condition
Theorem
condition
for a p p r o x i m a t e
can be o b t a i n e d
controllability
by d i f f e r e n t i a t i n g
similar
to the
(3.13).
3.16
We d e f i n e
U
= {u e U:
of U such that B U (3.2)
C
D
Bu e D
(A).
is a p p r o x i m a t e l y
U
controllable
-- { A n B U sp
(3.21)
(A) = ~ D(An)},
the
largest
subspace
is non e m p t y but n e e d n o t be d e n s e on
[O,tl]
n = 0,i,...,}
in U.
if
= Z
Proof S u p p o s e [3.2) is not , z , and all u s U
approximately
controllable,
then
for some non-
zero
, = 0 Z Z
>Z, Z = O
and h e n c e
Differentiating
O < t _< t I
yields
z, Z = 0
0 ~ t [ tl,
n = 0,i,2,...
For t = 0 this y i e l d s = O
*,AnBU> Z
this
z
and
(3.21)
theorem
seems
fact the c o n v e r s e
(3.21)
~et
3.17 Z = ~2 [_~,~3,
Az
d2z d~ 2
not hold.
to be a n a t u r a l to check
since
is s u f f i c i e n t
is in g e n e r a l
ing example.
Example
does
it m a y be d i f f i c u l t
the c o n d i t i o n
n = O,i,2,...
*Z
false.
extension U
of the rank
is not e a s i l y
b u t not n e c e s s a r y ,
To see this,
consider
derived. and in the
follow-
67
D(A)
dz d2z = {z: z, dE' dE=
z}
Then it is easy to show that 1 , (Ttz) (~) = ~
Suppose
the control
(~-n) 2
~ f
exp{-
4t
}z(n)d~
system is given by
= Az + blU 1 + b2u 2
where Ul,U 2 e L 2 [ O , t ~ , shown that if bl(E) C~(O,1)
with b2(E)
approximately
bl,b 2 c Z.
is nonzero, = bl(E-h),
controllable
on
Fattorini[12]
has compact [O,tl].
But bl(E)
outside
[Anbi i = 1,2
[-h,l]
Theorem
vanish to-
for h > O.
Hence
n = O,1,2 .... ]
of L2[-~,~]. if we specialize
For example:
3.18
If A is a b o u n d e d
operator
[O,tl] , then there exists sp
on Z and
(3.2) is exactly
an integer n, such that
[BU,ABU,... ,AnBU]
= Z
Proof From T h e o r e m
3.7 we know there exists
a 7 such that
. z*llLq[0,tl,U.] _> Jlz*IJz* YIIB.T(.) *
is
and bl(E-h)
Stronger results of the rank type can be obtained the semigroups.
have
(O,i) and b I e
h ~ O, then the mild solution
gether w i t h all their d e r i v a t i v e s
is a p r o p e r subspace
and Triggiani[44]
support on
But T t = e
A *t
=
~
t n "A*
7~ n-.' ~ n=o
~n, hence
controllable
on
68
YrlB*T~.)z*llLqlO,tl;U* I
n.'
n=o
_< y
1 n+l/ , Z llB*(A*)nz*ll . t I yn. (nq+l) n=o U
.E IIB*(A*)nz*II
Since the series
*
1 / tl tnq >~ " _ _ q n--Z-o[IB*(A*)nz*[IU* ~of (n.') dt
n+l
n=o
li
< y
Thus 7
~
o
. tI
/n: (nq+l) ~ _> llz*ll . Z
U *..
1
n
,
Z B [A*jnz * tl/n. converges
*
in U , there exists N
n=o
such that 7
]
Thus
N
Y
Now if y =
n+! . ! Z [IB* (A* )nz*II . t I q/nl (nq+l) q ~ ½1[z*II . n=N+l U Z 1
n+ =
Z IIB*(A*)nz*II . t I q/n: (nq+l) ~ --> ½11z*l] * n=o U Z
max n=O,l,...,N
n + ~1 2y t I
1 | (nq+l) q , we have
N
Z IIB*(A*)nz*H . _> IIz*[IZ* n=o U Using Corollary
3.5 we see that this is equivalent
to the statement
of
the theorem. In many problems of practical interest the null state plays an important role and so we introduce special definitions for systems which can achieve or approximately achieve the null state. Definition We say that
3.19
Exactly null controllable
on [O,tl]
(3.2) is exactly null controllable Range{G}
D Range{Ttl}
on [O,tl]
if
69
Definition We say that
3.20
Approximately
null c o n t r o l l a b l e
(3.2) is a p p r o x i m a t e l y
from C o r o l l a r y ectly we will
on
first introduce
[O,tl]
if
O Range{Ttl}
and exact null c o n t r o l l a b i l i t y
3.5 and T h e o r e m
duality r e l a t i o n s h i p s
[O,tl]
null controllable
Range{G}
Criteria for approximate
on
follow directly
3.6, but instead of e x p r e s s i n g four concepts
of o b s e r v a b i l i t y
to obtain expressions
these dirand exploit
for controllability.
In
order to do this we consider the system =
~x
,x(o)
=
x
(3.22)
o y = Cx
where
~
is the infinitesimal
group S t on a reflexive a reflexive
generator
of a strongly
Ban ach space X, and C e
Banach space.
(3.23)
The mild solution
of
continuous
~(X,Y)
semi-
where Y is also
(3.22) is
y = CStx o
and if x O e D ( ~ )
we may d i f f e r e n t i a t e
We define a map ~ ,
x(t)
= Stx O to obtain
(3.22).
by x
q[0,tl
Y]
:
o = cstx o
T~en w e say Definition We say that
3.21
Continuously
~3.23)
initially
is continuously
observable
initially
on
[O,tl]
observable
there exists y > O, such that ~ll~xll q[O,tl;Y]
Definition We say that
3.22
_> 11xllx
Initially o b s e r v a b l e
(3.23)
is initially
on
~ x e X.
[O,tl]
observable
ker(~)
= {0}
on
[O,tl]
if
on
[O,tl]
if
70 Definition
3.23
We say that
Continuously
(3.23)
finally o b s e r v a b l e
is c o n t i n u o u s l y
on
[O,tl]
finally o b s e r v a b l e
on
[O,tl]
if there
exists 7 > O, such that 711~XfILq[O,tl;Y]
Definition
3.24
We say that
~ lIstlxlfx
Finally o b s e r v a b l e
(3.23)
to reconstruct unique
[O,tl]
is finally o b s e r v a b l e ker(~)
The o b s e r v a t i o n
on
c
~ x e x
on
[O,tl]
if
ker(Stl)
p r o b l e m is concerned with w h e t h e r or not it is possible the initial
state or state at time t I of a system in a
fashion from the observations.
it is only possible
In many problems
to construct an optimal
of the s y s t e m is known.
Knowledge
of o p t i m i s a t i o n
control if the whole state
of the initial state enables
us to
obtain the w h o l e state at all times t ~ O, and this is why initial observability is important. distinguishable,
Definition
implies
x o = O, whereas
= o
continuous
of a continuous
on
[O,tl]
initial o b s e r v a b i l i t y
reconstruction
R ° : Range(~) In practice
state is
that is y(t)
existence
3.22 implies that the initial
it is not possible
finite time t I has elapsed.
operator
R
o
~ X , su(h that R o ~
to c o n s t r u c t
For times
implies the
= I.
the initial state until a
greater than tl, the whole state
can be d e t e r m i n e d by the state at time tl, so that in many control applications Definition
the weaker
concept of final o b s e r v a b i l i t y
3.24 implies
that the states
is more useful.
at time t I can be distinguished,
that is y(t) implies x(t I) = O, whereas existence
= 0
on
continuous
of a r e c o n s t r u c t i o n
operator
Rtl : Range(~)
[O,tl]
final o b s e r v a b i l i t y
implies
Rtl
÷ X, such t h a t
Rt'~l
= Stl
the
71 Now let us turn to the question
of duality.
For this we make the
dual identifications U =
Y
,
B
=
C
,
Tt
=
St,
Z
=
X
,
A
=
I~*
! + !q = i
P
Theorem
3.25
(a)
(3.23)
controllable (b)
(3.23)
is exactly (c) null
is initially on
is continuously
controllable
(3.23)
(3.23)
on
is finally
controllable
(d)
observable
on
[O,tl]
iff
(3.2)
is approximately
[O,t ~.
on
initially
observable
on
[O,tl]
if f (3.2)
[O,tl].
observable
on
[O,tl]
iff
(3.2)
is approximately
[O,tl].
is continuously
exactly null controllable
on
finally
observable
on [O,tl]
iff
(3.2)
is
[O,tl].
Proof We have already Parts
(c),(d)
proved
follow
in Theorem
from the construction
Theorem
3.7, and Corollary
Example
3.26
Consider
(a),(b)
3.5 and Theorem
the observation
problem
3.11 and 3.7 respectively.
of the dual of G as in 3.6.
on L 2 [O,i~
x t = x~ (3.24)
x(O,t)
= x(l,t)
= O, x(~,O)
= Xo(~)
1 y(t)
The dual problem
= [ o
b(~)x(~,t)d{
is the control zt(~,t)
problem
= z~(~,t)
+ b(~)u(t)
(3.25) z(O,t)
= z(l,t)
= O, z(~,O)
= Zo(~)
72
U s i n g the results of E x a m p l e
3.12 we k n o w that the m i l d s o l u t i o n of
(3.25) is a p p r o x i m a t e l y c o n t r o l l a b l e 1 f b(~)sin n~ o
iff
d~ ~ 0
¥ n = 1,2 ....
Hence by the d u a l i t y T h e o r e m 3.25 the same c o n d i t i o n s ensure that
(3.24)
is i n i t i a l l y observable. Example
3.27
In E x a m p l e
3.9
we s h o w e d that the c o n t r o l l e d d i f f u s i o n e q u a t i o n zt =
z~
+ u
(3.26) z(O,t)
= z(l,t)
= 0
is not e x a c t l y c o n t r o l l a b l e to all of z e L2[O,I] , u £ L2[O,tl;Z ] . The dual s y s t e m is x t = x~
(3.27)
x(O,t) y(t)
So c l e a r l y
= x(l,t)
= X(t)
(3.37) is finally o b s e r v a b l e
is a p p r o x i m a t e l y null c o n t r o l l a b l e . continuously
= 0
for any t I > O, and h e n c e
To see w h e t h e r or not
(3.26)
(3.27) is
finally o b s e r v a b l e , we n e e d to show there exists y such
that (3.28)
y2 ~ 2 ( l _ e - 2 n ~ 2 t l )
(
2nZz2
of
1
sin n~y z(y)dy
)2
~ ~ 2e-2na'~2tl It is easy to show that y~ = 1/t I s a t i s f i e s t i n u o u s l y finally o b s e r v a b l e controllable
l s i n n~y z (y)d
(3.28) and so
for any t I > 0 and
(3.27)
is con-
(3.26) is e x a c t l y null
for any t I > O.
One of the m o s t i m p o r t a n t p r a c t i c a l c o n s i d e r a t i o n s
in systems theory
is the d e s i g n of f e e d b a c k controls so that the c o n t r o l l e d s y s t e m is a s y m p t o t i c a l l y stable. Definition
3.28
Exponential
stabilizability
Let A be the i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y c o n t i n u o u s semi-
73 g r o u p T t on a B a n a c h If t h e r e e x i s t s uous
semigroup
space
D E
Z, a n d B e
~(Z,U)
There
(A,B)
dimensional
Banach
Example
,
and c o m p l e t e
spaces
with
the
complex
a strongly
space.
contin-
theory
stabilizable.
of s t a b i l i z a b i l i t y
in i n f i n i t e
following
implies
for f i n i t e
stabilizability.
dimensional
Banach
spaces
example.
3.29
L e t Z = £~, U = R I, a n d
for z = { ~ i , ~ 2 , . . .
Bu = { B l U , 8 2 u ....
Z n28~ n=l
£2,
< ~,
{81,82,...
}
1 = n~n
A(~n)
where
U is a B a n a c h
> 0
in t h a t c o n t r o l l a b i l i t y
is m u c h m o r e
as w e i l l u s t r a t e
Me
is s a i d to be e x p o n e n t i a l l y
is a v e r y n i c e
The s i t u a t i o n
where
T Dt w i t h IIT TI
then the p a i r
~(U,Z)
such that A+BD generates
}, 8n ~ O.
F o r any f e e d b a c k
} e £2
control
u =
d e £~ w e h a v e
~2 AZ + BU = { ~ l + 8 1 < d , z > £ 2 , ~
We w i l l a compact
show that 0 E ~(A+BD). operator,
however
for some n = { n l , H 2 , . . . , }
we give below
this
follows
the c o n t r a r y ,
since A+BD
a s i m p l e p r o o f by s h o w i n g
e £2 t h e r e a r e no s o l u t i o n s =
L e t us s u p p o s e
Actually
+82£2,...,}
is that
of
(A+BD)z.
then
~n = n q n - n S n < d ' z > £ ~
But
{n8 n}
exists
e i 2, and for s o m e q n , { n n n }
for t h e s e
and from Theorem
(3.29)
Un"
Thus A+BD
~ £2 a n d h e n c e
is not
invertible,
no such
(2.10)
O ~ sup{Rel,
I e ~(A+BD)}
D w h e r e T t is the s e m i g r o u D
generated
D I ~ inf{~:llTtl
b y A+BD.
{~n } e £2
a n d so O e ~ ( A + B D ) ,
M e~t}
74
Now if IIT~II _< Me ~t then from not
stabilizable.
approximately
(3.28) we must have e _> O, a n d so
Nevertheless
controllable
we
are a b l e
in f i n i t e
to s h o w
time.
For
that
this
we
the
is
(A,B)
system
is
require(Theorem
3.11) B Ttz
= O
t > O implies
z
= O
it *
But
Ttz
*
= {e
n
*
~n }, w h e r e
the
same
argument
8n~ n = O,
and h e n c e
able,
not
but
In g e n e r a l Indeed
*
*
*
= {~i,~2,...,}
!t B * Ttz * * = n Z= l S n en ~n
Therefore
Using
z
there
as
t _> O.
in P r o p o s i t i o n
~n = O.
Thus
the
3.13 w e a r e
system
able
to c o n c l u d e
is a p p r o x i m a t e l y
controll-
stabilizable. we
do n o t h a v e
are s t r o n g l y
equality
continuous
in
(3.29)
Hille-Phillips
semigroups
T t generated
[18]. by
an o p e r -
the
spectrum
a t o r A, w i t h sup{Rel,
and yet
IITtlI = e t
One m e t h o d of
I £ ~(A)}
= O
(see Z a b c z y k [ 5 3 ] ) .
we will
the g e n e r a t o r ,
use
to e x a m i n e
and we will
wish
stabilizability
to e x c l u d e
is v i a
the a b o v e
example,
so w e
define Definition
3.30
A strongly the
spectrum
Spectrum
continuous determined
determined semigroup
growth
sup Re
We
note
that
Triggiani alytic
this
and
for s o m e
immediate
that
then
assumption
the
this
Zabczyk
t > 0
consequence sup
T t with
assumption generator
A is s a i d
to s a t i s f y
if
~ (A) = l i m l°glITJ = ~o t+~ t
is a l w a y s
shown
semigroup,
is c o m p a c t An
has
growth
case w h e n
is a l s o has
the
shown
(see T r i g g i a n i of this
Re ~(A)
lJTtll ~ Me -~t
< -~
A is b o u n d e d , case w h e n
that
t ~ 0
same
some
is t h a t e > 0
an an-
is t r u e w h e n
[44~).
definition for
the
for t > t O .
A generates
if
Tt
75
We w i l l also n e e d to d e c o m p o s e the space Z. the p o r t i o n s ~u(A)
Let ~ > O and c o n s i d e r
and ~s(A~ of the s p e c t r u m of A c o n t a i n e d in the closed
half plane {l: Re I ~ -6} and open h a l f plane
{1: Re k < -6} r e s p e c t -
ively. au(A) = ~(A)
{l: Re I ~ -6}
~s(A)
{k: Re k < -6}
= o(A)
Then ~(A) = ~u(A)
U ~s (A) and since A is a g e n e r a t o r ~(A)
in the half plane
{l: Re I ~ ~o }.
Definition
3.31
Spectrum decomposition
If the set ~u(A)
assumption
is b o u n d e d and is s e p a r a t e d from the set ~s(A)
such a w a y that a rectifiable,
simple,
to e n c l o s e an open set c o n t a i n i n g ~u(A) exterior,
is c o n t a i n e d
in
closed curve can be drawn so as in its i n t e r i o r and ~s(A)
in its
then A is said to satisfy the s p e c t r u m d e c o m p o s i t i o n assumption.
If A s a t i s f i e s the s p e c t r u m d e c o m p o s i t i o n a s s u m p t i o n then Kato
[2.7]
has shown that (a) the o p e r a t o r A may be d e c o m p o s e d a c c o r d i n g to the d e c o m p o s i t i o n Z
=
Zu
+
Zs
of the space, m e a n i n g PD(A) c D(A), AZ s C
Zs, AZ u C
w h e r e P is the p r o j e c t i o n on Z, Z u = PZ, Z s =
1 (II-A)-idl ¢ P = 2hi fF
Zu
(I-P)Z, and
~(Z)
and F is a curve s u r r o u n d i n g ~(Au). (b) O(As) = ~s(A),
~(A u) = ~u(A)
where A s and A u are the r e s t r i c t i o n s of A on Z s and Z u r e s p e c t i v e I y . (c)
A is b o u n d e d on Z u
(d)
P and
(I-P) commute w i t h A
PA c AP (e)
,
(I-P)A c A(I-P)
T t c o m m u t e s w i t h P and
(I-P)
76
(e)
is n o t p r o v e d
note
P is b o u n d e d
in K a t o b u t
is a c o n s e q u e n c e
PR(I,A) Hence
= R(I,A)P
from
(d).
To s e e
this we
I e p(A)
~o
R(I,A)
we
of
and
= I e-ltTt z dt o
obtain O = PR(I,A)z
Then
by the
- R(I,A)Pz
uniqueness
= S e-lt(PTtz-TtPz)dt o
of the
Laplace
Transform
we
have
PT t = TtP We now
apply
this
decomposition
(3.30)
= Az
F r o m Pz = Z u , ( I - P ) z applying
P and
(I-P)
= z s, to
+ Bu,
denoting
strongly
u (a) Tt,
Ts t are
by A u and A s . tinuous
semigroup
strongly
Actually
analytic
= z o.
t h a t P~ = Z u , ( I - P ) z
= A U zU + PBu
+
s the by T u t and T t
continuous
system
zOU
(I-P)Bu
restrictions
Tt,
= Pz °
Zos
=
(I-P)z o
to Z u a n d
continuous
semigroups
on Z u a n d
semigroup
Anu t n / n !
= n=o
(3.31)
also
apply
P,
z(t)
and
Zs of the
(I-P)
to the mild
solution
t = Ttz o + S Tt_sBU(s)ds o
to o b t a i n t Z u ( t ) = T ut Z o u
Zs generated
on Z u a n d T u t is a u n i f o r m l y
At
could
a n d so
then
A u is b o u n d e d
T tu = e u
One
= is,
we have
~s = A s Z s
Also
z(O)
it f o l l o w s
(3.29)
ZU
to the dynamical
+ [ T tu _ s P B U (s) d s O
con-
77 t s Zos + f T t_s(I-P)Bu(s) s Zs(t ) = T t ds o If we assume
that the semigroup T ts satisfies
growth assumption,
the s p e c t r u m d e t e r m i n e d
we have s ll _< Ke -6t llTt
This d e c o m p o s i t i o n ity of
(3.31).
provides
the clue for analyzing
We note that the p r o j e c t i o n
onto Z is n a t u r a l l y stabils The idea therefore is to stabilize
izable by taking the zero control. the p r o j e c t i o n Theorem
the s t a b i l i z a b i l -
onto Z u w i t h o u t
upsetting
the stability properties
on Z s.
3.32
(A,B) is e x p o n e n t i a l l y
stabilizable
(a) the g e n e r a t o r A satisfies
on a Banach space
Z if
the s p e c t r u m d e c o m p o s i t i o n
(b) the semigroup T ts satisfies
the s p e c t r u m d e t e r m i n e d
assumption,
growth assump-
tion, (c) the p r o j e c t i o n
onto Z u is e x p o n e n t i a l l y
control u = DuZ u, D u e
stabilizable
by a feedback
~(Zu,U).
Proof We show that
(A,B) is stabilizable
by means of the control D =
(Zu,Z s) + U. On Z
u
we have
(Au+PBDu)t Zu(t)
and from the assumption
= e
Zou
(c) there exists C,p > O such that
IIZu(t)ll < ce-PtIlZou[l
Hence for u = DuZu, we have llu(t)ll < CIIDuHe-PtllZou[[
Feeding the control u = DuZ u onto Zs, yields t [Izs(t) II -< Ke-~tl] Sos ]l + MllZoull f e - ~ ( t - s ) e - p s d s o
(Du,O):
?8
=
where M
=
e 6t e-Ptl Ke-~tllZosll + MIJZoullL ~ J
KcJ](I-P)BIJIIDull'
Hence the response of the system
li
(t)(I = I l l s ( t )
+
(3.31) to u = Dz is
u(t)ll
IMFe-6t-e < Ke-~tllZosJl + ~ L -pt]
< KIII-Ple-~tjJzolJ + k
L
~
> + ce-P t IIZoulJ
J + ce-Pt
)
IIPIIjlzOjl
and this completes the proof. Corollary 3.33 If the
of Theorem 3.32 hold except that
conditions
(c) is replaced
by (a) the space Z u is finite dimensional (b) the projection onto Z u is controllable then again we have
(A,B) is exponentially
stabilizable.
Exanple 3.34 Let A be self adjoint with compact resolvent, then there are at most a finite number of non-negative eigenvalues, say J. Hence Zo is finite dimensional and using Example 3.12 we can easily show that the condition for stabilizability of (A,B) with B = (bl,b2,...,bn) , bj e H is
rank
[
H :
.....
LH ,
H i
= rj
j = 1 ..... J.
,H
There are not many results which connect controllability with stabilizability without any constraints on the semigroup or its generator. The major general result is the following:
79
T h e o r e m 3.35 If the m i l d solution of on a Hilbert
(3.2) is exactly null c o n t r o l l a b l e
space H w i t h controls
Hilbert space,
then
(A,B)
in
[O,tl]
u e L 2 [ O , t l ; U ] where U is also a
is e x p o n e n t i a l l y
stabilizable.
Proof The simplest proof of this assumes knowledge quadratic
control problems w h i c h we d e v e l o p
will anticipate
these results here.
null controllable that z(t I) = O.
in
[O,tl]
time linear
Since the mild solution
there exists
= O for t > t I.
We
is exactly
a control u e L 2 [ O , t l ; U ] such
We now play this control on
for t > tl, then z(t)
of infinite
in the next chapter.
[O,tl] and the zero control
Hence
co
f [b(s)llH2
J(u) =
+ llu(s)II~]ds <
o
We show in the next chapter that this condition existence J(u).
of a unique o p t i m a l
Moreover
this control
control
u
is sufficient
~ L2[O,~;U]
for the
which minimizes
is of the feedback type, being d e t e r m i n e d
via the unique b o u n d e d operator
solution
of an algebraic
Riccati equation.
Hence i m p l e m e n t i n g
feedback
control ensures
the e x i s t e n c e
of a D e
~(H,U),
this optimal such that co
(3.32)
f o
(HTDZolI~ + HDTDZo]I~)dt _< K21]ZoI[2
D is g e n e r a t e d by A+BD. where T t
Now
(3.32)
implies
f llT~Zoll~ dt ~ K211Zoll2 o and the proof is completed by the f o l l o w i n g
lemma.
Lemma 3.36 If T t is a strongly
continuous
that
semigroup on a Hilbert
oo
f IfTtZol1 dt <_ WTI%II 2 o
then there exists constants
M,~ > O, such that lITtl[ _< Me-~t
v Zo
£ H
space H, such
80 Proof Assume
IITtll ! ~e~t for some M,~
[ O, then since
1 - ~2mt 2~
IITtZoII~ = I
te_2~ ~
IITt~oll~ as
o ! fte-2~llTeII211Tt_~Zoll 2 d~ o t o
! ~II~oll ~ we have
IITt~olf ~ ~II~oll for
some y i n d e p e n d e n t of t.
Thus
IITtll ! and
t
t
tIITtZoJl2 = f IITtZollz d~ <_ / IFT ll211Tt_ zoll2 d~ o
o
< ~K~II ~oll~ Hence
IITtzoll!
t~
IlZoll
¥ zo
But this implies that llTtll < 1 for t s u f f i c i e n t l y
H.
large.
Then ~0 = lim
logllTtl I
< O
t and so there exists M,~ > O such that
iITtrl ! Me-~t Example
3.37
In E x a m p l e
3.8 we showed that the c o n t r o l l e d w a v e e q u a t i o n was exactly
c o n t r o l l a b l e on
[O,tl]
for any t I > O, and hence it is also e x a c t l y null
c o n t r o l l a b l e and so s t a b i l i z a b l e .
In E x a m p l e
3.25 we showed that the
c o n t r o l l e d d i f f u s i o n e q u a t i o n was exactly null c o n t r o l l a b l e and h e n c e
81
stabilizable.
Actually
the system is stable.
this is obvious However,
in this example since for u = 0
it is easy to extend the example
sider the case where the uncontrolled
to con-
system is unstable.
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CHAPTER QUADRATIC
4.1
Regulator We suppose
4
COST CONTROL
PROBLEM
problem that our system is described
by the following
input-output
relationship (4.1)
t z(t) = Tt_toZ ° + ~ Tt_sBU(s)ds o
where H, U are real Hilbert is a semigroup continuous
on
The cost (4.2)
~to,tl]
J(U;to,Zo)
Our control
~
with
this is called
of proof is to construct
to show that they converge the following
the state
F k(t)
= -R-IB*Qk_ l(t)
~(t)
= M + F~(t)RFk(t)
;
is the perturbation
With the choice of control
(4.7)
z(t)
and
u* e L2(to,tl;U)
Z(tl)
problem",
the
close to the zero state. of feedback
controls
control.
sequence
= Fk(t)z(t)
We also consider
is
e > O
the "regulator
to the optimal
Uk(t)
Zk(t)
control
a sequence
F o (t) = O
Qk(t) h = U ~ ( t l , t ) G U k ( t l , t ) h
(4.6)
(4.1)
and nonnegative
and some
is to find an optimal
being to bring
where Uk(t,s)
is strongly
J(U;to,Zo).
Our method
(4.5)
the system
V u ~ U
motivation
(4.4)
z(t)
u c L~(O,tl;U).
are self adJoint
~ flu 112
In finite dimensions
Consider
z O e H, and {Tt; t ~ O}
From lemma 2.24,
= +ftlI+}ds to
and R e ~(U)
problem
which minimizes
associated
O ~ t o ~ t ~ tl <
B e ~(U,H), A.
for all inputs
functional
where M,G e ~(H) (4.3)
spaces,
on H with generator
;
t1 U~ (s,t)Mk (s) Uk (s,t)h d s + I t
of T t by BFk(t).
given by
(4.4),
(4.1) becomes
= Uk(t,to)Z o the controlled = Uk(t,to)Zo
version
of
(4.6)
t + t~ Uk(t,s)Bu(s) o
ds
and
86
for some
u ~ L 2(O,tl;u).
Now a series of lemmas are established which demonstrate the convergence of the feedback controls (4.4) to the optimal solution of the problem Lemma 4.1
=
+ ~tl{ t
-2}
ds
where z is given by (4.7). Proof This is by direct substitution of (4.7) and (4.5) into the above expression and for simplicity we verify the result in two parts. (a) Suppose we only consider terms in G : t L.H.S. = +2
t G / Uk(tl,S)BU(s)ds > to
Uk(tl,S)BU(s)ds > o
since G is self adjoint and using the semigroup property of Uk(t,s) (see Definition 2.32). Similarly for the other side, t 1
R.H.S. = + 2
+<~ to
Uk~tl,S)B~(s)ds,
Gt] Uk(tl,S)B~(s)ds> o
G~ tl Uk(tl,S)BU(s)ds > to
t 1
- 2 f t
GUk(tl,S)B~(s)>
ds
t 1 s -
2 ~ t
f to
d~ ds
=L.H.S. (b) Suppose we only consider terms in M and R : t1 L.H.S. = f t
ds
+ 2ftl/t t to
d@ ds
87
÷
f
t
f
t
f
t1
<Sk(S,s)Bu(s),
Mk(e)Uk(~,8)Bu(8)>
de d8 ds
to to t using the semigroup
property
of Uk(t,s).
t 1
R.H.S.
= f t
~(S)Uk(S,to)Zo
> ds
+ 2 ftlfs as as t to tl s s + f d~ d8 ds t t t o o t 1 t 1
- 2 f
t
f s
~(~)Uk(S,s)Bu(s)>
- 2 ftlftlfS
t
s
as ds
, ~(~)Uk(S,s)B~(s)>
d8 as ds
to
= L.H.S. interchanging the order of integration
in the last two terms.
Lemma 4.2 The cost for
(4.1) with the feedback
control
(4.4) is
J(Uk;to,Z O) = For fixed z ° ~ H and fixed to< tl, is monotonically asing in k and _<
decre-
¥ z o e H.
Proof (a) Let u = O in lemma 4.1, then
Qk(to)Z(to)>
= < z ( t I) , Gz (t 1 ) >
=
+ f tl (
GZ(tl)>
+ ftl
Mz(s)>
+
Mz(s)>
+
~(s)z(s)>
ds
RFk(S)Z(S)>)
ds
to =
+ f tl (
t = J(Uk;to,Z O) from
o
(4.2)
RUk(S)>)
ds
88 (b)
Let u(t) = (-Fk(t)
valent to Zk+l(t) Substituting
tl{
+ / to
+ Fk+l(t) )z (t)
(4.7) is equi-
Qk(to)Zo > =
in lem~a 4.1, we obtain GZ(tl)>
RFk(S)Z(S)>
t1 + / to
+ 2
-
+
Then
= Uk+l(t,to)Z °
for this ~(t)
=
in (4.7).
t1 + I { to
t/tl{
(s) z (s) , RFk+l(S)Z(S)>
ds
Qk(S)BFk(S)Z(S)> % Qk(S)BFk+I(S)Z(S)> } ds
+ } ds
+
RFk(S)Z(S)>
o
2}ds
=
+ ftl<(Fk+l(S) to R(Fk+l(S)
- Fk(S)) z (s),
- Fk(S)) z(s)> ds
That is, =
+ ftl
y(s)> ds
where y(s) = (Fk+ 1 - Fk(S))z(s) From
(4.3) we have the inequality
_>
for all k and t O < t I. Lemma 4.3 Qk(t) of (4.5) converge strongly as k÷ ~ to a self adjoint operator Q(t) e ~{H) for each t e [to,tl] and Q(t) satisfies the integral equation t4.8~
Q(t)h = U~(tl,t)GU(tlt)h
where U(t,s)
is the perturbed
t~e perturbation
+ /tlu~(s,t)(M+Q(s)BR-IB~Q(s))U(s,t)h t mild evolution
operator
of T t by -BR-IB~Q(t).
Q~t~ is strongly
continuous
in t on [to,tl].
corresponding
ds to
89 Proof From
(4.5),
Qk(t)
is a self adJoint
for each t and from lemma 4.2, Qk(t)
and nonnegative
is decreasing
operator
in ~(H)
and uniformly
bound-
ed in norm:
llQk(t)ll
=
sup llZoll=l
<
sup
[IZoll=l < Hence for each t, Qk(t) operator ~(t)
Q(t)
£ ~(H)
also converge
a constant
C t
converges
and Q(t)
strongly
_< c on
to a self adjoint nonnegative
[to,tl].
From
as k + ~ to-R-iB*Q(t)
strongly
(4.5), Fk(t)
and
and M+Q(t)BR-IB*Q(t)
respectively. We now proceed +BFk(t),
to show that Uk(t,s),
converges
by -BR-IB*Q(t).
strongly
as
From Theorem
the perturbation
k --'- ~ t o
U(t,s),
the
of T t by
perturbation
of
Tt
2.33, we have t
(Uk(t,s)-U(t,s))h
= - f Tt_pB(Fk(P)Uk(P,s)-R-1B*Q(p)U(p,s))h s t ll(Uk(t,s)-U(t,s))hll < f llTt_pBFk(P)llllUk(P,s)h-U(p,s)hll dp s
and
t + f llTt_pBllll(Fk(P)-R-1B*Q(p))U(p,s)hll s The second convergence Gronwall's
term converges theorem
to z e r o
and since Fk(P)
+ U(t,s)
strongly
We may now apply the Lebesque k ÷ ~ in the expression (4.8).
is uniformly
bounded
dp
dominated
in k, using
lemma Uk(t,s)
The strong
of U(tl,t)
as k + ~ by the Lebesque
dp
on
for Qk(t)
continuity
as
dominated in
k + convergence
(4.5) to prove
of Q(t)
follows
theorem
that Q(t)
from the strong
to let satisfies continuity
[to,tl].
Lemma 4.4 The optimal
control which minimizes
J (U;to,Z o) is the feedback
trol u(t) and the minimum
cost is
= -R-iB*Q(t)z(t)
.
con-
90
Proof Consider an arbitrary controlled system is
admissible
control u e L2(to,tl;U)
t z(t) = T t _ t o Z ° + ~ T t _ s B U ( s )
so that the
ds
o Since U(t,s)
is a perturbation
of T t by -BR-IB*Q(t),we
z~tl = U(t,to)Z ° + ~tu(t,s)Bu(s) to
also have
as
where ~(t) = u[t) + R-iB~Q(t)z(t) Applying
lemma 4.1, for t = t o , we obtain
= + ~ttl{ o +
=
B Q(s)z(s)>
GZ(tl)> + ~
-
2
Q(s)Bu(s)>}ds
Mz(s)> + < u ( s ) ,
Ru(sl>
ds
o
- /'tl<~(s), NOW from
(4.8)
= + f t
tl{
R~(s)> ds
to +
o RuW(s)> } ds
and so we have J(u*;to,Z O) = J(u;to,Zo)
_< J(U;to,Zo)
- ftl
Ru(s)> ds
from lemma 4.2
(4.3)
But this inequality holds for arbitrary ~ £ L2(to,tl;U) optimal.
and so u ~ is
Lemma 4.5 Q(t) is the unique solution of uous self adJoint operators
(4.8) in the class of strongly contin-
in ~(H).
Proof Let Q(t) be the strong limit of the sequence Qk(t) of
(4.4) and
91
suppose that P(t) is another solution of (4.8). [4.9)
P(t)h = U~(tl,t)GUo(tl,t)h
+ f t
t1
That is
Uo(S,t)(M+P(s)BR-IB~P(s)) Uo(s,t)h ds
where Uo(t,s)
is the perturbation of T t by - BR-IB~P(t)
Consider the system t (4.10) z(t) = Tt_toZo + t~ Tt_sBU(s) o
ds
for some u e L2(to,£1;U) Using the defining equation for Uo(t,s) from Theorem 2.33, substituting u for u in (4.10) we obtain o t (4.11) z(t) = Uo(t,to)Z O + t~ Uo(t,s)B~o(S) ds o where Uo(t) = Uo(t) + R-lB*P(t)z(t) NOW by lemma 4.1, for P(t) given by (4.9) and z(t) by
(4.11) we h a ~
t1 (4.12) = + [ t
(
+
- 2)ds
The cost for control Uo(t) is given by t1 J(Uo;to,Z O) = + I ( + ) to
tl{
= + f tO
+
+ - 2Ids Hence from (4.12) (4.~3)
J(uo;to'Zo)
Since U(t,s) equivalent to (4.14)
= + ftl
R~o(S)>ds
is the perturbation of T t by -BR-IB*Q(t),
t z(t) = U(t,to)Z o + f U(t,s)Bu(s) to
ds
(4.10) is
ds
92
where u(t) = u(t)+R'iB*Q(t)z[t). So arguing as above for Q(t) given by we can show that (4.15)
J(U;to,Zo)
=
Now if u(t) = -R-iB*P(t)z(t)
(4.8) and z(t) given by
+ ftl
(4.14)
R~(s)> ds
in (4.10), we have
H(t) = R-iB*(Q(t)-P(t))z(t)
in (4.14)
or z(t) = Uo(t,to)Z O For this particular
choice of u, by lemma 4.2 and (4.15), we obtain t 44.16) J(U;to,Z O) = + f l d t =
Similarly, and
choosing
p(to)zo
>
Uo(t) = -R-IB*Q(t)
in
(4.10),
and using lemma 4.2
44.13) we obtain
(4.17) J(Uo;to,Z O) = + ftl dt = Adding
(4.16) and t 1
f to and
From
(4.17), we obtain
llR-½B*(P(t)-e(t))U(t,to)Zoll2dt
= 0
f tl IIR-½B* (Q(t)-P(t))Uo(t,to)Zoll2dt = O t o (4°3), we deduce that B*Q(t)U O(t,t O)z ° = B*P(t)U O(t,t O)z O on [to,tl]
and
B*Q(t)U (t,to)Z O
which from Theorem
= B*Q (t)U (t,to) z O
2.33 implies
that U(t,t O) = Uo(t,to),
and so in fact
B*Q(t)U(t,to)Z ° = B*P(t)U(t,to)Z O = B*P(t)Uo(t,to)Z ° Thus from
(4.9) and
(P(t)-Q(t))h
(4.8), = ftluo(S,t)Pis)BR-iB*P(s)U O~s,t)h ds t - ]tlu* (s,t)Q(s)BR-IB*Q(s)U(s,t)h t =
0
ds
93
In fact we can show that Q(t) Riccati equation
analogous
is the unique solution of a differential
to the finite dimensional
Riccati equation
(1.6). Lemma 4.6 Q(t)
~s the unique solution of the following
equation
in the class of strongly
~(H) with ~4.18)
d
differentiable
+
= Q(t I) = G
continuous
inner product Riccati
self adjoint operators
in
for h,k c D (A). +
+ <Mh,k>
on [to,tl]
where h,k ~ D(A).
Proof (a)
To show that the unique solution of
form the inner product of
by term using the following property ~4.19)
(4.8) satisfies
(4.18) we
(4.8) with h,k £ D(A) and differentiate of the perturbed
U(s,t)h = -U(s,t) (A-BR-IB*Q(t)~h
a.e.
evolution
term
operator.
for h ~ D(A) and s>t
(see 2.43). It is routine to verify that the formal differentiation and so we shall only Justify this differentiation Consider g(t) =
then and using
]tl<MU(s,t)h, t
ds, k>
U(s,t)k>
yields
(4.18),
for a typical term.
for h,k e D(A)
ds
(4.19) d_~_(t) = -<Mh,k> dt = -<Mh,k>
+ f t
tl
<MU(s,t)h,
U(s,t)k>
ds
- ftl<Mu(s,t) (A-BR-IB*Q(t))h, t - ftl<Mu(s,t)h, t
since all terms are integrable.
U(s,t)k>
ds
U(t,s) (A-BR-IB*Q(t))k>
ds
Hence
94
d_g_(t) = <Mh,k>-< (A-BR-IB*Q(t))h, [tlu* (s,t)MU(s,t)k ds> dt t -
(A-BR-IB*Q(t))k>
is a closed operator.
(b) Uniqueness
Let Ql(t) and Q2(t) be two solutions of (4.18), then writing P(t) = Ql(t)-Q2(t), it is readily verified that (4.20) ~t = -<(A-LQ l(t))h, P ( t ) k >
-
(A-LQ l ( t ) ) k >
- and d (4.21) ~
= -< (A-LQ2(t) )h, P(t)k> - +
where L = BR-IB * If F(t)h = ftlui[s,t)P~s)LP(S)Ul£S,t)h t
ds, where Ul(t,s)
is the pertur-
bation of T t by -LQI(t) , then as in (a), for h,k E D(A), we may differentiate to obtain (4.22) ~t = - -
(A-LQI (t) )k> -<(A-LQ l(t))h, F(t)k>
F(t I) = O Assuming for the moment that
(4.22) has a unique solution, we have
F(t) = P(t) and = ftl t for all h e H.
Similarly using
(4.21), with Q2 perturbations, we obtain
the inequality < 0 for all h e H and hence P(t) = 0
ds _> 0
95 (c) Uniqueness
of
(4.22)
Let S(t) = T~_sF(t)Tt_ s then for h,k e D(A) d d-~
is differentiable
in t and
=
+
-
ATt_sk>
- <(A-LQl(t))Tt_sh , F (t) Tt_sk>
=
+
So
~ttl< (QI(P )L*F (P) + F (p)LQ 1 (P))Tp_sh,
=-
T
p-s
h>dP
for all h e H, since D(A) = H. Letting s+t, we obtain t1
= -f t
<(QI(P)L*F(~)
+ F(p)LQl(P)Tp_th,
Tp_th>
dp
Since F(t) is self adjoint
llF(t)ll--
sup
ll
llh,=l <
Or
Q(t) of (4.23)
f t
t1
CIJF(p)JJlJhJJ2 dp, for some positive constant C
llF(tlIJ <_ C ftljJF(p)Jl dp t
so by Gronwall's Corollary
sup JJhJJ=l
inequality,
HF(t)JJ = O and
(4.22) has a unique
4.7 (4.8) or (4.18)
is also uniquely
Q(t)h = U*(tl,t)GTtl_th
determined
+ t~tlu*(s't)MTs-th
by ds
or
solution.
96
t1 (4.24)
Q(t)h = T ~ l _ t G U ( t l , t ) h
(4.25)
Q(t)h = T~l_tGTtl_th
+ tf
T~_tMU(s,t)h
ds
t1 T~_ t [M-Q (s) BR- IB*Q (s) ] Ts_th ds
+
t
Proof (4.23)-(4.25)
4.2
are established
with h,k e D(A)
by showing
yields
that differentiating
(4.18).
Examples
Example If holds
4.8
Finite
Tt= e At
dimensional
where A is an n × n matrix,
for all h,k
e R n and our theory
al regulator
problem
Example
Parabolic
4.9
For our system semigroup
2.40.
H = R n, U = R TM, then
agrees with the finite
(4.18)
dimension-
in §i. systems
(4.1) we suppose
generated
of Example
systems
that the semigroup
by the self adjoint
Then the Riccati
T t is the analytic
operator A on a Hilbert
equation
(4.18)
space H
can be decomposed
as follows: Suppose
that it is possible
to express
Q(t)
by
Q(t)h = iZ,j=o Z qiJ (t)$j For simplicity
we assume
¥ h e H
that all the eigenvalues
of A are unrepeated;
then since A~j = ~j~j
,
(4.18) becomes
qij + (~i+~j)qiJ
+ ~M~i'~j>
- ~ij = O
(4.25) qij(tl) where uij
=
=
Z Z qikqjr k,r=o
Of course we need to justify and we do this
~4.26)
that such an expansion
for the normalised
heat equation
I
zt = z ~ + u(t,~)
!
z~(O,t)
= O = z~(l,t)
;
for Q(t)
of Example
z(~,O)
= ZO(~)
converges
1.2.
97
where we seek
to m i n i m i z e
1
J(u)
= ] Z a (tl,~)d~ o
Then for ~i =
/2 cos ~i~
+ ] O
t I (fl z
; i=i,2,...
qij" - z ~ ( i 2 + j 2 ) q i J
2(t
,~) + u 2 (t,~)d~)
dt
o ;
G o = i , (4.25)
+ gij - k=lT. q i k q j k
becomes
= 0
(4.27) I qij (tl) = diJ
Now q i j ( t ) = O we have
is a s o l u t i o n
qij (t)50 I
ii - 2 ~ 2 i 2 q i i qii(tl)
In this
to
(4.27)
and since
the s o l u t i o n
for i~j and + 1 - q~i = O
= 1
case we can solve
explicitly
for qii(t),
obtaining
a i(l-b i) - b i ( l - a i ) e -ei(t-tl) qii (t) =
where
Hence
(l-bi)
~. = 2 ~ i l
-
(l-ai)e-e i(t-t I)
~ +
a.l = - n2i2
-~i
~ + 1
b i =-
+~i
~ +
~2i2
[qii(t) J is u n i f o r m l y
bounded
in i and t and
oo
Q(t)h is well
defined. u(t)
where
=
=
Z qii (t)#i<- ,~i > i=o Similarly 7 ui(t)~i i=O
we can e x p a n d ,
ui(t ) = - qii(t)zi(t)
z(t)
=
u(t)
and z(t)
~ zi(t)~ i , i=O
by
is unique,
98 Example
4.10
Consider
the controlled
(4.28)
wave equation
Ztt = Z~E + u(t,~) z(O,t)
= Z(itt)
= 0 , Z(~,O)
= Zo(~)
, Zt(~,O)
= Zl(~)
with the cost functional 1 It11 (~z~(~,t) + u 2(~,t)) a(u) = % I (z~(~,t I) + z~(~,tl))d ~ + o o o dE dt
(4.29)
We can express defined
(4.28) as an abstract
in Example
2.41.
system on the Hilbert
(oI
(4.30)
w(t) -kZt(t) / = Tt Zl
+ of
(4.31)
J(u) = ½<w(t l),w(tl)>
Tt_ s
space d~
u(s) ds
t1 + ] o
(%<Mw(t),w(t)>
+ H) dt
M = <00 O> and the inner product onD~is given by
where H = L2(O,I),
<w,~>~ =
<.wE, 'wE-I>H+
<w2'~2>H
wI for w =
We suppose that Q(t) given by (t)
Q2 (t)~
\ Q3 (t)
Q4 (t)/
Q (t) = |/ Q I
For h =
and k =
h2
(4.18) can be expressed
k2
e D~),
(4.18) reduces
to
d d--t <(Qlh1+Q2h2)E'k~>
d + d-t
+
+
+
+
+
%
(4.32)
=
as
99
Ql(tl) = %I = Q4 (tl)
;
Q2 (tl) = O = Q3(tl)
where all the inner products are in H. We now introduce the following expansions for Qi(t); i=1,2,3,4. 0o
Q1 (t) =
Z Z eij¢i<" '~j>H i,j=l
Q2 (t)
iEjZ=l~ij*i < '*j>H
(4.33) Q3(t) = i,j= Z Z 17.19 O i <',~j>H Q4(t) = i~j__Zlnij*i<" *j>H where
~i(~) = /2 sin wi~ .
Since Q(t) is self adjoint in ~6 , we deduce that uij (t) = eji(t)
;
~ij (t) = nji(t)
(4.34) ~2k28ki = Yik Substituting h I= O = k I and h 2= ~m' k2= ~n and Qi(t) given by (4.33) in (4.32), we obtain 6nm + ~2n28nm + ~ m 2 ~ m n
+ %~mn =
(4.35)
E Bim~in i=l
nnm(tl ) = ½~nm Substituting k s= O = h 2 and h I= ~r' kl= ~s and Qi(t) given by (4.33) in (4.32), we obtain w2S2~sr - ~2r2~r s - w2S2Ys r = i~iYirYis (4.36)
{ ~sr(tl ) = ½~sr
in
Substituting h == O = k 2 and h 2= ~m' kl = ~n and Qi(t) given by (4.33) (4.32), we obtain
(4.37)
I ~2n2~nmsmn(tl)=+ o~2m2Umn " ~2n2nnm =
By inspection,
a solution is
E i=lDimYin
8nm = Ymn = O, ~mn = O for m ~ n, = %. Since (4.18) has a unique solution nnm = 0 for m ~ n and Umm = nnlm it must be
100
O(t)
i(1o)
=
O and so the optimal
control
is
I
u(t,~) = -
and the m i n i m u m
zt(t,~)
cost
1
is
f o
Example
4.11
Delay equations
We consider
the controlled
linear
N
~(t) = AoX(t)
+ ~
delay equation
~x(t+ei)
;
t+6 i > 0
~h(t+Si)
;
t+% i < O
Ai
i=l
o (8) ~ x ( t + e ) d 8 r + J Aol -b ~h(t+o)d0
(4.38)
;
t+8 _> O
;
t+@ < 0
+ Bu(t)
x (O) = h (O) where
-b <_ ON _< 8N_ 1 _< ''' <_ 81 < O ;
Aol e C(-b,O;
~(Rn)),
as in Example
that we wish to choose
a control
cost functional
is m i n i m i z e d
(4.39)
= <X(tl),GX(tl)>Rn
b > O , Ao,Ai;
2.42
i=l,... ,N e ~(Rn),
and B e ~(Rm,Rn).
We suppose
u E L 2 (O,tl;R m) so that the following
t1
where
J(u)
G,M e ~(R n) are self adjoint
strictly
positive
As in Example
(<x(s),Mx(S)>R n+Rm)ds
and positive,
and R £ ~(R m) is
and self adJoint. 2.42 we reformulate
(4.40)
z = A z + B u
(4.41)
J(u)
where
+ of
;
z(O)
(4.38)
and
(4.39)
on ~ 2 ( - b , O ; R n)
= h t1
= ~t~+
B, G and M
are degenerate BU ;
( B u)(8)
{
o
;
maps onto j~2,
;
=
Gh(O);O ;
for
u g Rm
for
h e ~ 2 ( - b , O ; R n)
for
h e ~ 2 (-b,O;R n)
o
0~O = O
= O
given by
0~o
m-,(o)~e
O
(~2+Rm)ds
e = o
=
( M h) (.0) =
( ~ h)(0)
f o
8 ~0
101
and A is given by
N
o
A°h(O)+i=l Z Aih(ei)+ -b~ A°l(8)h(8)d8
;
8
=O
~h (8) = dh(8) de
8/0
with domain, D(~) = J~2(-b,O;R n) NAC2(-b,O;Rn). We can decompose the solution Q(t) of (4.18) as follows Q(t)
=
/
Qll(t)
QI2 ( t ) )
\
Q21 (t)
Q22 (t)
where Qll(t)e
~(Rn),
Q22(t) e ~(L2(-b,O;Rn))
Q21(t) E ~ (Rn,L2(-b,O;R n))
and
Ql2(t) = Q21(t)
Substituting this into (4.18), we obtain Qll(t)+ AoQll(t)+ Qll(t)Ao+ M = QII(t)BR (4.42)
B Qll(t)
- 02(t)- Q~(t) QII (tl) = G
Now define Q21(t,@) by Q21(t,8)x = (Q21(t)x) (8)
for x e R n
and Q2(t) E ~(R n) by Q2(t)x = Q21(t,O)x . Substituting these into (4.18) yields the following equation for Q21(t,e) a ~ +
(4.43)
Q21(t'8)---Q21(t'8)+~8 Q21(t'e)A°+ Agl(e)Qll(t) N-I i=l
,
AiQll(t)~(8-Si)- Q21(t,8)BR-1B*Qll(t)+ Q22(t,e,O)= 0
Q21(t,-b)= ANQII(t);
Q21(tl,e)= O;
Q21(t,O)= 0
Q21(t,8) is piecewise absolutely continuous in e with jumps at 8 i of i
height AiQll(t) ;
i=l,2,...,N-i .
102
we define Q22(t,8,e)
by
(Q22(t)h) (8) =
o f Q22(t,8,~)h(~)de -b
and derive the following equatioI~s for Q22 (t,8) Bt Q22 (t'@'~) + ~~
Q22 (t,e,~) + ~~
,
(4.44)
=
A'ol(~)Q21 (t'@) + Q21(t'~)AoI(8) +
N-1 Z Q21(t,~)Ai6(e-Si) i=l
Q22(t,-b,8) Q22(t,O,@)
Q22 (t'8'~) N-I + i=lZA'iQ21(t,@)~(~-@i)
- Q21(t,~)BR
= ~Q21(t,@),
Q22(t,s,-b)
B Q21(t,8)
= Q21(t,8)~
= O = Q22(t,~,O)
Q22 (tl'8'~)= O Q22(t,~,8)
Q22(t,~,8) l
*
AiQ21(t,8)
= Q22(t,8,~)
is piecewise continuous
at ~=8i;
i=l ..... N-I
For proofs and explicit ational methods 4.3
in e and 8 with jumps of height
(resp. Q21(t,8)A i at ~=ei; i=l .... ,N-l).
solutions of these see
[ii] and for comput-
for solving these Riccati equations
see [12] and
[2].
Tracking problem
Often we do not wish to bring the system to the origin, but rather to a preassigned final state or even to follow a preassigned trajectory as closely as possible. ing problem"
In finite dimensions
and its mathematical
formulation
this is called the "trackis to minimize the per-
formance index (4.45)
J(U;to,Zo)
= + f t
over all u ~ L2(to,tl;U),
t1
(
+ )dp
o where r(t) is a given continuous
H-valued
function on [to,tl]. Again our method of proof is to construct a sequence of controls but this time as a combination of feedback and open loop
103
u k(t) = -F k(t)z(t) - R-iB*Sk_ l(t) where Sk(t) : -U~(tl,t)Gr(tl) - I tl U~(p,t) [Mr(p) t (4.46)
+ (Qk(P)-Qk_I(P))BR-IB*sk_I(P)]dP s o(t) = o
and Uk, Fk, Qk are defined by (4.5) as before. Since Uk(t,s) is a quasi-evolution operator, as in the proof of Lemma 4.6 we can show that sk(t) also satisfies the differential equation ~t <sk(t)'h> = - <Sk(t), (A-BR-1B*Qk_l(t))h> + <Mr(t),h>
+ < (Qk (t)-Qk-i (t))BR-IB*sk_I (t) ,h>
(4'.47)
for all h e D(A)
sk(t I) = Gr(t I)
We now consider the sequence of control problems t z(t) = Tt_toZo + ~t Tt-sB(Uk(S)+~(s))ds o t Uk(t,to)Z b + ~ Uk(t'p)B(u(p)-R-1B*Sk-I(P))dP o As a consequence of Lemma 4.1, we deduce that (4.48)
(4.49)
= + "jtl dp t
-2 Itldp t By direct substitution from (4.47) and (4.48) and interchanging the order of integration as in Lemma 4.1, we can also prove that (4.50)
= - t 1
- f t
- ftl dp t
d~
104
Adding for
the
(4.49) and cost
(4.50) with t=t
of the
control
o
and ~ = O, we obtain an expression
u k. t1
(4.51)
Jo(Uk;to,Zo)
= +
+ f t
dp
o
2- 2~ tl < B R - 1 B * S k _ 1 (2) ,S k (p)> dp to t 1 + f <Sk_z(p),BR-lB*Sk_l(p)> dp t o
-
Similarly adding
(4.49) and
(4.50) with t= t
o
and rearranging,
we
- )
dp
obtain (4.52)
Jo(Uk+~;to'Zo)
- Jo(Uk;to'Zo) ftl t
where
=
(
o
y(p) = R-iB*~Qk(p)-Qk_l(p))z(p)
+ Sk(P)
- Sk_l(p) ~
If u(Q) = -R-iB*((Qk(p)-Qk_l(p))z(p)+sk(P)-Sk_l(p)), that Jo(Uk;to,Zo) converges
is decreasing
in k.
then
(4.52) shows
We can show that Jo(Uk;to,Zo)
as k + ~, from Lemma 4.3 and the following result.
Lemma 4.12 (a) (4.53)
Sk(t) converges
strongly as k ÷ ~ to
s (t) = -U* (tl,t)Gr(t I) - f tl U* (p,t)Mr (p) dp t
and satisfies i d
<s
the differential (t),h> = -<s
(t)
equation. (A-BR-IB*Q(t))h>
+ <Mr(t),h>
(4.54) s (t I) = Gr(tl)
(b)
If furthermore,
T~G and T~M map H to D (A*) and
t1
I I llA*T~hlldt < (4"55) }t;t. "I to
for all h e H
105
then s (t) is the unique s (t)=(4.56)
s
solution
of the evolution
(A*-Q(t)BR-IB*)s
equation
(t) + Mr(t)
(tl) = Gr(tl)
.P.roof (a)
From Lemma
in k, and so from
4.3 Qk(p)
and Uk(P,t)
(4.46) we deduce
that sk(t)
in k and t on
[to,tl].
gence theorem
to let k + ~ in
obtained
(4.53) by differentiating
from
evolution (b)
property (4.53)
Let Y(t,p) U(t,p).
are uniformly
(2.43)
bounded
dominated
thus obtaining <s
in norm
is also uniformly
We can now apply the Lebesgue (4.46),
bounded
(4.53).
conver(4.54)
is
(t),h> and using the quasi
of U(p,t).
is the mild solution
= U*(tl-O,tl-t)
denote
Now by Corollary
4.7,
of
(4.56) by Definition
the dual mild evolution
2.23.
operator
to
(4.24), t
T~_pO(P)h
= T~l_tGU(tl,P)h
+ I P
T~_tMU(u,O)h
Hence since T~G and T~M map H to D(A*), we see that T~_pQ(p)
: H + D(A*)
du
A* is closed
and
(4.55)
holds,
and t
A*T~_pQ(p)h
Moreover
(4.55)
by Theorem operator (4.55)
= A*T~l_tGU(t,p)h
implies
that A*T~_pQ(p)h
2.37, A*-Q(t)BR-IB * generates
and by Corollary
is sufficient
Now let k + ~ in
+ / P
A*T~_tMU(~,0)h
is integrable an almost
(4.50)
(4.52)
to have a unique
- Jo(U which
;to,Zo)
letting k ~ ~ in Jo(U
control. (4.51),
;to,Zo)
strong
strong
and
solution.
= ~ o
_ _
dp
shows that
(t) = -R-IB*Q(t)z (t)-R-iB*s
is the m i n i m i z i n g
and so
to obtain tI
Jo(U~+U;to,Zo)
u
(to,t)
strong evolution
2.39 that A*-Q(t)BR-IB * is almost
for
for all ~e L 2 (to,tl;U),
on
d~
(t)
The cost for this control
is obtained
by
namely = + -2
(to)>- ftl<s t o
+ftldp t -i o (p),BR B*s (p)> dp
106
In fact this result limiting
versions
sequential problem
of
approach
because
and because
the optimal
could be obtained
(4.49)
and
(4.50).
more directly However,
of its analogy
to our proof
it is a useful numerical
procedure
the the
for the regulator for calculating
control.
Finally we remark homogeneous
that a quadratic
cost control
problem
for the in-
system t w(t)
t
= Tt_to z o + ~
Tt_pBU(p)
dp + t~
o where
by proving
we have outlined
g e L2(to,tl;H)
Tt_pg(p)
dp
o
can be reduced
to the tracking
problem
considered
a b o v e , by l e t t i n g t z(t) = w(t)-r(t),
Example
4.13
r(t)
= [ t
Tt_pg(p)
dp
o
Heat equation
Let us consider
the system given by
in Example
(4.26)
4.9, but this
time with the cost functional 1 (4.57)
J(u)
= f o
tI (z(tl,X)-sin~;x)2dx
1
+ f o tI
1
÷fo We have
already
Expressing
s
found Q(t)
(t) =
and so it remains
Z si(t)~ i, i=O
(z (t,x)-sin~x) 2dx
f o
f o
to solve
(4.56) becomes
si (t) = +(n2i2+qii(t))si(t)
+ ~il
(4.58) s I (tl)= 6ii Hence
si(t)= O, Sl(t)
and the optimal u
= 2et-tl-i
control
is
(t) = - i=iZ qii(t)zi(t)#i
u 2 (t,x) dx dt
- sl(t)#l
(4.56)
for s=(t).
107
4.4
The infinite We c o n s i d e r
time q u a d r a t i c
cost control p r o b l e m
(4.1) on the infinite
interval
t (4.1)
z(t) = Tt_toZ o + ~
Tt_sBU(s)
ds
o with
the
cost
(4.59)
functional
J = ( U ; t o , Z o) = ~ (~Z(S),Mz(s)> t o
w h e r e we i m p o s e
the
same assumptions
and in addition we assume that ists a K e ~ ( H , U )
such
that
the
+ )ds
on Tt,
B,
u,
M, R a n d
(A,B) is optimizable; feedback
control
Ul(t)
z ° as
before
that is, there ex= -Kz(t)
yields
a finite cost and m o r e o v e r
~=(Ul;to,Z o) ~ ~11zoll~ for some constant ~ > O and all z
e H. We note that if (A,B) is stabilo A g a i n the p r o b l e m is to find an optimal
izable then it is optimizable. , control u e LZ(to,~;U) w h i c h m i n i m i z e s proof is similar to that in §4.1. of feedback
J
(U;to,Z o) and the m e t h o d of
We consider
the following sequence
controls
~4.60)
uk(t)
= FkZ (t)
Fk = -R-1B*Q~_I;
(4.61)
F1 = K
M k = M + FkRF k os Tkh Qk h = f TkM. s K s ds o
k is where T t
the
that
is
(4.61)
owing
perturbation well-defined
of
Tt
and
so
lemmas, which are analogues
b y BF k . first
It
is
not
we e s t a b l i s h
of Lemmas
immediately
clear
this
foll-
in
the
4.1 and 4.2.
Lemma 4.13 tI tl t1 (-2)ds = [ t tI where Qk (t) is given by t 1
(4.62)
Qk
tl
(t)h = f t
Tk_t~Tk_th
ds
and
z(t) --Ttk_toZo+
t k
f ~ B~Cs) t _-s O
ds
108 Proof As for Lemma well-defined Lemma
4.1, where we have
implicitly
assumed
that MkT ~ are
for each k.
4.14
t1 Qk (t) converges
strongly
The cost of the feedback
as tl~ ~ to Qk given by
control
(4.60)
(4.61)
for each k.
is given by
J(Uk;to,Z o) = and Qk is m o n o t o n i c a l l y
decreasing
in k and bounded
above
and below.
Proof The proof
is by induction,
so first we suppose
that k = i.
From
(4.62) we have t1
(4.63)
= ftl
M.T 1 ± s-t-a
z ,Zo> ds o
tl-a
= is increasing tl ftl = t O
(t)Zo>
in t I for fixed t o and Zo' and (
+ )ds
J(Ul;to,Z O) ~ ~llZoll2 by the
o p t i m i z a b i l i t y assumption. t1 So Q1 (to) is an increasing sequence
ors which
are bounded
ly to Ql(to) of to,
above,
as tl+ ~.
which
Finally
of bounded self adjoint operatt that Q l(t ) converges strong-
proves
(4.63)
shows
that Ql(to)
is independent
and so we have = J(Ul;to,Zo)
Suppose
~ ellZo]l2
t1 now that Qk (to) converges
strongly
to Qk as tl+ ~, and that
~ ~llZoll2 for k ~ n. Consider
the feedback
(4.64)
tl Un+l(t)
If we let
u(t)
= -
control
R-IB *Atl (t) z (t) ~n
t1 = Un+l(t)-
Un(t)
in Lemma
4.13, we have
109
t1 t1 ( + = ~ t tI t1 + + <~(s),RUn+l(S)>) ftl =
ds
tl t1 +
ds
t tI + ~ t Thus from
tI t1
ds
(4.3) and letting t = to, we obtain ~tl
t! tl t1 + ds ~ <2o,O n (to)Zo >
t (4.65)
o co
< < eUZoll 2 by our induction assumption. Now by Fatou's
J
lemma, we have the inequality ._n+l
(Un+l;to,Zo)
.
_n+l Zo>d s
= ~ <~S_toZo,Nn+lWs-t t O
o
tI tI ) lim i n f ( ftl(< z (s),Mz(s)> + )ds tl~ \t O
co
_< So from
<_ ~llZoll2
(4.62) we can substitute e~
=
from
(4.65).
t = t o and t I = ~ to obtain
-n+l . _n+l t] ds o
~llZolj2
_< <
Arguing as for k = l, we deduce that Qn+l(to) hence
is independent of to, and
J (Un+l;to,Z O) = ~ ~ ellZoll2
Lemma 4.15 Q~ converge strongly to Q~ as k + ~ and QZ satisfies
c466)
= f
ds
O
where T t
is
the
perturbation
of
Tt
by -BR
B*Q .
110
The optimal control u*(t) (4.67)
is given by
u*(t) = -R-IB*Q~z(t)
Proof As in Lemma 4.3, Qk converges
strongly
to Q~ using Lemma 4.13 and
for finite tl, tl .tl ~* ~ -I ~ Q~ (t)h = ] Ts_t(M+Q BR B*Q=)Ts_t h ds t
(4.68)
Combining
these yields
(4.66), using the fact that Q~ is time invariant,
it can be proved that the optimal
control
law is given by
(4.67) as in
Lemma 4.4. In order to show that Q~ is the unique solution Riccati equation, er assumptions
as for the finite dimensional
that
(A,B) and
of an algebraic
case, we need the strong-
~A*,M ½) are stabilizable.
Lemma 4.16 Suppose positive (4.69)
(A*,M ½) is stabilizable
Q c ~.~H) and K e ~(H,U), <2(Q(A-BK)
Then A-BK generates
and there exists a self adjoint, such that
+ M + K*RK)h,h>
an exponentially
<_ O
for all h ~ D(A)
stable semigroup.
Proof Since
(A*,M %) is stabilizable,
A*-M½S generates
an exponentially
A-S*M % generates
an exponentially
there exists S ~ ~(H) stable semlgroup,
A-BK = A-S*M½+(S*M½-BK) and by our perturbation
formula
(2.31),
t
c4
Tt h. S
ds
o where T~ is the semigroup Let h = T~k in
generated by A-BK.
(4.69) to obtain
such that
or equivalently 1 say. But stable semigroup Tt,
that
111
d ds
(½ ) + IIM½T~kli2 + IIR½KT~kli 2
0
for all s > O and k E D(A). Since Q ~ 0 and R is invertible, we deduce that co
f iiM~T~kli ~ d~
<
o
(4.71)
for all h e H. oo
f llET~kil ~ d~
<
o
From (4.70), we have the inequality t ]IT~hll
i[T~hl[ <_ 21iT~hll2 + const
Since T tz is exponentially f iiT~hli2 dt < ~ o
of ,IT~_s,i2ds
stable and
of(llM½T~hil2 +
s)
,IKT~hli2)d
(4.71) holds,
for all h e H.
By Lemma 3.36, this implies that T t2 is an exponentially
stable semigroup.
Corollary 4.17 If (A*,M ½) and (A,B) are stabilizable, exponentially stable semigroup.
then A - B R - I B * <
generates
an
Theorem 4.18 If (A*,M ½) and (A,B) are stabilizable, then Q: given by unique solution of the algebraic Riccati equation (4.72)
+ + <Mh,k> =
for h,k e D(A).
B*Q h,k>
(4.66)
is the
112
Proof (a) Existence Let Q: be given by
(4.66),
oo
writing
--1
and consider ~
B*Q~)k>
co
+ < (A-BR-IB*Q~)h,Q
k>
÷
A ~ = A-BR -i B*Q~ = ] (< (M+Q BR o
B*Q~)Tsh,A
Tsk>
+ . . ) ds =
s
from
T k> ds
(4.66)
for h,k
E D(A),
o
= - oo
since,
by Corollary
Q: given by
4.17,
(4.66)
T t is exmonentially -
is a solution
of
stable with generator
A ~o o
;
(4.72).
(b) Uniqueness Suppose
that
(4.72) has two solutions
2
+
and by Lemma
4.16, with K = R-IB*Q,
ially
semigroup
stable
Writing (4.73)
K
o
= R-IB*p,
Then
from
+
A-BR-IB*Q
generates
(4.72) we have
= 0 the exponent-
TQ
the following
<(2P(A-BKo)
P and Q.
identity
is readily verified.
+ KoRK ° + M)h,h>
= <(2Q(A-BK)
+ K*RK + M)h,h>
+
2< (P-Q)(A-BK)h,h>
+ 2< (K-Ko)*R(K-Ko)h,h> but P and Q both satisfy (4.74)
(4.72) with h = k and so
<2 (P-Q) (A-BK)h,h>
Since T Q is exponentially
+ < (K-Ko)*R(K-Ko)h,h>
stable,
for h e D(A), we have
oo
< (P-Q)h,h>
= -S d~ o
= O
s
ds
113
= -fm ds o by
Q = ½f ds
(4.74).
O
Since D(A)
is dense
Another
for the filtering Lemma
of our results
problem
Q ~ P.
which has important
implications
of §6 is
4.19
t1 (A*,M ½) are stabilizable and Q (t) denotes the solution t1 (4.8) (or(4.18)), then Q (t) converges strongly to Q~, the unique
If of
in H, P ~ Q and similarly
consequence
(A,B) and
solution
of
(4.72)
as t I ~ ~ for all t ~ O.
Proof t1 (a) Let Qo (to) be the solution
of
(4.8)
for G = O.
Then since
tl tonically
is the optimal
increasing
is bounded
above on
as t I + ~ for each From Lemmas
[to,~).
we have Qo
(to) is monot1 is stabilizable, Qo (to)
(A,B) t1 Hence Qo (to) converges
strongly
fixed t O .
4.13 and 4.14 we recall
and we can also deduce
where
[to,tl],
in t I and since
tl tl ÷ ~ Qk (to)h
(4.75)
cost on
that
~h k ~ ~ ~h Qk " Q~
Yh
c H
that
tI ~ Q~ tl T~ ~(M+Q~BR-IB*Q~)T~h~ w ~ Qk (to)h k/__~__, ih = f o the convergences
are in the strong
t1 Since Qo (to) is optimal,
sense.
we have
tI t1 Qo (to) ~ Q~
and letting t I
, we obtain
Po(to ) ~ Q~
for all t o ~ O
ds
to Po(to)
114
But applying Fatou's
lemma to
(4.75), we have
< t P o where U p ( s , t )
is the perturbation
o f T t by - B R - 1 B * P o ( t ) .
But the left side is the cost of the feedback control u = -R-IB*p
o (t) z (t)
on [O, ~) and since Q~ is optimal on [O,~),
Hence
Q~ < Po(to)
for all t o _> O
Q~ = Po(to)
for all t o ~ O
t1 (b) Let Q (tO ) be the solution of is optimal
(4.76)
(4.8) for G ~ O, then since Q tl (t o )
tI +
t1 >
(to)Zo,Zo>
> Hence
tI O < <(Q
t1 (to) - Qo (to))Zo'Zo >
tl <(Q 1 _ Qto(to))Zo'Zo>
~ +
÷ O as tl + ~ by
(4.75) and Corollary tI
So
<(Q
4.17
t1 (to)-Q O (to))Zo,Zo > ~ 0 as t I ~
tI t1 which implies that Q (to) + Qo (to) strongly as tl÷ eralised Schwarz inequality
I[BII 2 ~ So from
(a), Qtl(t o) converges
Example 4.20
using the gen-
for B ~ ~(H)
strongly to Q~ as tl+
The Heat Equation
Let us return to the heat equation in Example 4.9. We know that for ~ . 2 6 ~ the system operator is self adjoint and generates a stable semi-
115
group.
Hence the infinite time problem is well posed and the algebraic
Riccati equation has the unique solution Q = i=l
l'
(/~i ~ +
-
~2i2 )~i<-,~i >
The optimal control is given by co
u(t) =
Z
- ( / ~ i ~ + i' - ~2i2)zi(t)~i
i=o i:o
where z(t) =
Z zi~t)~i , and the minimum cost is i=o
Z
( / ~ i ~ + i - ~2i2)() a
i=o
Example 4.21
Wave equation
In the Example 4.10 we have considered it is obvious that the infinite time problem is well posed since the optimal control and minimum cost are independent of the time interval.
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CHAPTER
STOCHASTIC
5.1
Abstract
and
X-valued
random respect
expectation
remark
that
theory.
variable
is a m a p
~ + X which
to p m e a s u r e .
the
these
Definition The
variable
is s t r o n g l y
measurits
is
we
a complete will
u induces
a measure
Pu on
~(x),
the
£ A}
probability
be w i t h
introduce
the
for
Hilbert
A e
~(X)
space. space-valued
random
variables
following.
operator
of an H - v a l u e d
random
variable
u e L2(~,p;H)
is
by
u
o v e
Coy{u}
The
u:
of
~(H)
= E{(u-E{u})
is d e f i n e d
(5.1)
identities
E{}
I Strong
and w e a k
=
are
all u , v
e H by
; h e H.
is s y m m e t r i c , p o s i t i v e
following
o (u-E{u})}
for
(u 0 v ) h = u < v , h > Then
the
If u is i n t e g r a b l e 2 on ~ w e d e f i n e
= p{~:u(m)
Cov{u} where
use
5.2
covariance
defined
X a Banach
shall
f~u dp
=
random
of o u r w o r k
for
space,
then we
probability
of X, b y v i r t u e
(X, ~ ( X ) , p u)
and
spaces,
by
sets
Most
probability
Hilbert
EQUATIONS
[2],[7],[8],[10]
abstract
Pu(A)
and
TheorY
is a c o m p l e t e
separable
E{u}
Borel
DIFFERENTIAL
5.1
able I with
We
STOCHASTIC
and E s t i m a t i o n
(~,@,p)
standard
Definition An
that
H,K are real
following
AND
Probability
Suppose space
PROCESSES
5
and n u c l e a r .
readily
trace
measurability
verified
Cov{u}
concepts
are
identical
for s e p a r a b l e
Hilbert 2 The
integration
used
here
is in the
sense
of B o c h n e r .
spaces.
(5.2)
E{<Bu,u>}
A Hilbert
space-valued
its c h a r a c t e r i s t i c Definition
= trace(H
random
Coy{u})
variable
for B 6 ~(H).
is a l s o
uniquely
specified
by
functional.
5.3
If u e LI (~,p;H) functional
Xu:
with
induced
H + R
Xu(h)
measure
is d e f i n e d
Pu'
then
its
characteristic
by
= E{exp(i)} = f exp(i)dPu(k)
for all
h e H
.
H
Of
special
Definition
basis The
random
variable
measures
or
random
variables.
all
u is G a u s s i a n
i, w h e r e
if < u , e i >
is a r e a l
{ e l } i = O is a c o m p l e t e
Gaussian
orthonormal
for H.
corresponding
Theorem
measure
the
theorem
is c a l l e d from
Gaussian
and w e h a v e
the
following
[iO].
5.5
A measure
~ is G a u s s i a n
if a n d o n l y
if its
characteristic
function
X~
operator
on H.
form x~(h)
where
Gaussian
variable
for
characterization
has
are
5.4
An H-valued random
interest
~ e H and
- ½}
