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0 'Iv «.,->
~ >
is the scalar product of U = L2(r 1).
E
2 L (0, T;.U)
Hence ( 6.51)
A priori estimates for the approximating dual arc PE follow as in preceding cases, i.e. one multiplies (6.50) by PE and sgn pE (more precisely by ~\(p /\ E) where ~~ is given by (3.66) and integrate on [O,t] and Q, respectively. We get Ip (t)1 2 E
+
JT lip (t)11 2 1 dt 0 E H (0)
J ISE(y)p Idxdt -< C.
+
E
Q
£:
In particular, this implies that {pI} is bounded in L1(0,T;L 1(0)) +L 2(0,T;V I ) £: c L1(0,T~L1(0) + VI) c L1(0,T~(Hs(0) n V)I) (by the Sobolev imbedding theorem) where s > N/2. Thus selecting a subsequence we may assume that PE
+
p weakly in L2(0,T;V), weak star in Loo(O,T~H)
(6.52)
n
and by the Helly theorem PE (t)
+
p(t) strongly in (Hs(O) n V)' for all t E [O,TJ.
(6.53)
n
Since the injection of V into H = L2(0) is compact, we have by Lemma 5.1, Chapter 1 in [50J Ip)t)-p(t) 12 -< c.
242
~llp
(t)-p(t) £:
111l+C(~) v.
lip (t)-p(t) t
II
.
(H s (0)
n V)
I
for all A > O.
Together with (6.52) and (6.53), this implies that
PE ~ p strongly in L2(Q)
(6.54)
n
and by (6.52), (6.53) we conclude that P£ (t)
pet) weakly in L2(O) for all t E [O,T].
+
n
Now by Lemma 5.4 and (6.49), (6.50) QgEn(t'Y
~~
)
E
E ag(t,y*) weak star in
Loo(O,T~H)
n
PE (T)
peT) E -d$O(y*(T»
+
weakly in H.
n
E
Since {{3 n(YE )PE } is bounded i.n Ll(Q) and therefore weak star compact in n
00
n
00
(L (Q»* there exists w E (L (Q»* and a generalized subsequence {A} such tha t •A
8 (YA)PA
Thus taking Pt
E
+-
wweak star in
+
{En}
(6.55)
(L (Q»* .
= A to the limit in (6.50) we see that p satisfies the equations
6p-W
E
dg(t,y*) a.e. in Q
peT)
+
8p 8v
~ ap = O'ln ~1' p --
+
co
C
8$0(y*(T»
(6.56)
3 0 in 0
o·ln
~
~2'
8p -- O'ln ~3' ~ dV
(6.57)
Similarly, letting E = £n tend to zero in (6.51) we get
P E dh(u*) a.e. in ]O,T[. Equations (6.56), (6.57) must of course be understood in the sense of distributions. To conclude the proof it remains to verify (6.39), (6.42). To this end we use inequalities (5.64), (5.65) in the proof of Theorem 5.2 together with relations (6.48), (6.49), (6.54) to get En p£ 8 (YE)
n
n
1
+
p(f O- y:t- 6y *) = 0 strongly in L (Q)
(6.58) 243
(6.59) By Egorov's theorem, for each n > 0 there exists Q c Q such that m(~Q )
~p
-ag(t,y*))y* = o.
REMARK 6.3 Theorems 6.3 remains valid if g:[O,TJ of the form (boundary observation)
(6.39)' 1
x H
(n)
+
R is a function
g(t,y) = Jf1 gO(t,y(o))do where gO:[O,TJ
x
R + R+ is measurable in t, locally Lipschitz in y and
Ivygo(t,y) 1< C(1 + Iyl) a.e. t E ]O,T[, y E R. The optimality system (6.39) to (6.42) has in this case the following form: Pt +
~p
p(T) +
= 0 in [y* a~O(y*(T)) 3
>
OJ, p = 0 in [y* = 0; fO f OJ
0 in n
This follows by an easy adaptation of the proof, taking in (6.44) gE(t,y) = Jf1
g~(t,y(o))do
and noting that if y + Y1 weakly in L2 (0,T;V) n W1,2([0,T];V') then the 2 sequence of traces {~EJ is compact in L (L1)' REMARK 6.4 For the approximation of an optimal control we may use the following control process: 244
Minimize
(6.60) on all (y,u) subject to (6.45).
Here h is the smooth convex function defined as in Section 1.5. Arguing as above we see that for some subsequence En + 0, {u E } is weakly 2 'V n convergent in L (O,T~U) to an optimal control u of problem (6.1). For numerical algorithms we refer the reader to [79J. Example Consider an absorbing tissue ~ (Example 4.1) bution of oxygen concentration at moment t. concentration, it is required to arrive at a by choosing an adequate oxygen flux u on the constraints~ 0 < u(o,t) < p a.e. (o,t) E r.
and denote by y(x,t) the distriIf YO(x) > 0 is the initial concentration yO(x) at t = T boundary r, subject to the
To this end, consider the following problem:. Minimize T
J~
!yo(x)-y(x,T)ldx
+.
fo
( 6.61)
h(u(t))dt
on all (y,u) subject to (see (4.52)) Yt - 6y
>-
-1, y
>-
Yt - 6y = -1 y(x,O)
=
0
a . e. in Q
a.e. in [y
YO(x), x E ~; ~~
=
>
u in r
OJ =
r
x
JO,T[.
In cost functional (6.61) the first term penalizes the difference between the actual and the desired concentration. Here h:L 2(r) + R is the function defined by h(u) = 0 if 0 < u < p a.e. in r, h(u) = +
00
otherwise.
Then (see Proposition 1.9)
245
8h(u)
{w
E
L2(r)~ w = 0 a.e. in [0
in [u
<
u
<
pJ, w ~ 0 a.e.
= pJ, w < 0 a.e. in [u = OJ}.
By Theorem 6.3, if (y*,u*) is an optimal pair of problem (6.61) then there oo 2 exists p E BV([O,T];.(Hs(Il))') n L (0,T;.H 1(1l)) n L (0,T;.L 2(1l)) which satisfies the system (Pt
+
6p)a
0 a.e. in [y*
>
OJ
~ = 0 in L, p = 0 in [y* = OJ p(x,T) E sgn (yo(x)-y*(x,T)), x E ~ while the optimal control is giv.en by
o u*(a,t)
if p(a,t)
<
0
p if p(a,t)
>
O.
={
Thus if yO f y*(T) it follows that p f 0 a.e. in L and so the optimal control u* is bang-bang. Now we shall present a variant of problem (6.36) in the case where the control functions u are in the space
and yo = 0, fO = - AO. As seen in Section 4.3, the v.ariational inequality (6.37) has in this case a unique solution in W1,oo([0,TJ;H) n w1•2([0,TJ;V) and it is equivalent to the one-phase Stefan problem (4.68) to (4.70). We associate with this equation the control problem: Minimize
f: g(t,y(t))dt
+ X(u) +
¢O(y(T))
(6.62)
on all u E U and y E Wl,oo([O,TJ~H) n Wl,2([0,TJ~V.) subject to (6.37).
Here g [O,TJ x H + R+, ~O:H + R+ satisfy assumptions (vi) in Section 5.1 and X:U + R is a lower semicontinuous convex function. Let (y*,u*) be an optimal pair for problem (6.62). 246
To obtain the optimality
system we shall proceed as in the proof of Theorem 6.3. siders the control problem with pay-off T
Jo gE:(t,y(t))dt + X(u) and state system (6.45). 2
(T
Iluli U :: J
o
+
1 2 21Iu-u*llu
+
Namely, one con-
E: ¢O(y(T))
Here 11·11 u is the natural norm of
U,
i.e.
frl/u'(t,a)1 2dtda
where u' is the derivative of u:[O,T] -+ L2(r 1). If (y E: ,u E: ) is an optimal pair for this problem then by the above argument it follOi.Js via Proposition 4.6 that
If PE:
E
uE
-+
u* strongly in U
YE
-+
y* strongly in W'
1 co
([O~T];H)
1 2
n W'
([O,T]~V).
W(O,T;V) is the solution to (6.50) then we have X1(uE:'v)
+
hl (u~-u*')v'da
J~l
dt -
PE: vda dt >- 0 'Iv E U. (6.63)
Relations (6.52) to (6.55) remain valid in this case. Hence, arguing as in the above proof, we infer that there exists p E L2(O,T~V) n Lco(O,T~H) n BV([O,T]~(Hs(n) n V)I) which satisfies (6.39) to (6.42). Now letting E tend to zero in (6.63) we get X'(u*,v.) -
f~l
pvdadt >- 0 'Iv
E
U.
(6.64)
We identify the dual space U1 of U with L2(~1) under the pairing
JLl
q(a,t)v'(a,t)da dt 'Iv E U, q E U1.
Then (6.64) yields
f: where dX:U
p(a,s)ds - dx(u*)(a,t) 3 0 a.e. (G,t) E "1 +
(6.65)
U1 is the subdifferential of X.
Summarizing, we have
247
THEOREM 6.4
Let (y*,u*) be an optimal paip in ppoblem (6.62).
Then thepe
exists
whic~
satisfies (6.39) to (6.42) and (6.65).
§6.3 Boundary control of free boundary
problems~
the Dirichlet problem
We now treat the boundary control of the variational inequality (y'(t),y(t)-z)
+
a(y(t),y(t)-z) <
J~
fO(x,t)(y(x,t)-z(x))dx U
a. e. t E J 0, T[ V z E K
(6.66)
U
y(O) = Yo' y(t) E K for t E [O,TJ where a:H1(~) U
K
H1(~)
x
= {z
+
R is defined as in problem (6.37) and
E H1(~)~ z
= u in r 1, z = 0 in f2' z>
0 a.e. in ~}.
Here ~ = ~1~2 where ~1' ~2 are two open and bounded subsets of RN having sufficiently smooth boundaries f1 and f2' respectively (Figure 6.1).
Figure 6.1 As seen in Section 4.3, (6.66) represents the variational formulation of the obstacle problem Yt - 6y = fO
a.e. in [y
>
0]
Y> 0, Y - 6y > fO a.e. in Q = ~ t
y(x,O)
248
= YO(x) a.e. x
E
~
x
]O,T[
y = u in El = r 1
x
JO,T[, y = 0 in E2 = f2
x
]O,T[.
In the following we shall assume that fO E Lq(Q);yo E w~-(2/q)(D)' YO > 0 a.e. in ~, YO = 0 in f2
(6.67)
u E w2-(1/q),1-(lJ2q)([1)~ u> 0 a.e. in [1' u(o,O)
= YO(o)
for
G
E f1
(6.68)
where q > 2 is a fixed real number. As seen in Theorem 4.7, under these assumptions the variational inequality (6.66) has a unique solution y E W~,l(Q). Now we are ready to formulate the optimal control problem associated with state equation (6.66).
f:
g(t,y(t»dt
+ ¢(u) +
¢O(y(T»
(6.69)
2 ,1(Q) and u E W2-(1/q),1-(1/2q)(E ) subject to (6.64) and to on all y E W q 1 constraints (6.68). Here g~[O,T] x L2(D) + R~ and ~0~L2(n) thesis (vi) in Section 5.1 and 2-(1/q),1-(1/2q)([ ) ¢·w . q 1
+
+
R~ are assumed to satisfy hypo-
R is a lower semicontinuous, convex func-
tion satisfying the condition u> 0 a.e. in [1' u(o,O) = YO(o) a.e.
° E f1
for all u E D(¢). (6.70)
This assumption allows us to incorporate the control constraints (6.68) into the definition of ¢. We note also that if ¢ satisfies a growth condition of the form ¢(u) > C1 Ilull~ + C2
VU E
D(¢)
where C > 0 and II· Ilq is the norm of Xq = W~-(1/q),1-(1/2q)(l:1)' then 1 problem (6.69) has at least one solution (this follows as in Proposition 5.1, 249
taking into account that by virtue of estimate (4.111) the map u pact from X to Lq(Q)).
+
y is com-
q
The maximum principle for this problem has the following form: THEOREM 6.5 Under assumptions (6.6?), (6.68), (6.70) where q > (N + 2)/2. 2 let (y*,u*) E W ,1(Q) x X be an arbitrary optimal pair of problem (6.69). q 2 q co 2 2 1 Then there exist ~ E L (Q) and pEL (O,T~L (~)) n L (O,T~HO(~)) n BV([O,T]; H-s(~)), where s > N/2, such that
Pt
+
(Pt p(T)
6P E (Lco(Q))*, ~~ E X~ +
~)y*
6p -
+
~
= 0,
d¢o(y*(T))
3
(6.71)
E dg(t,y*) a.e. in Q.
0 in
(6.72)
~.
(6.73)
p(Yt - 6y*-f O) = 0 a.e. in Q
(6.74)
- ~~ E dp(U*) in 2: 1 •
(6.75)
In (6.71) we have denoted by X* the dual space of Xq , and in (6.72) the q 2 1 product (Pt + 6p-~)y* makes sense because y* E Wq ' (Q) c C(Q) for q > (N+2)/2. Proof We follow the method of Section 6.2. optimal control problem:
We start with the approximating
Minimize
T
HS(y,u) = fa gS(t,y(t))dt
+
¢(u)
+
~ Ilu-u*ll~
+
¢~(Y(T))
(6.76)
2
over all (y,u) E W ,1(Q) x X subject to q
Yt - 6y
+
q
SS(y) = fO in Q
y(x,O) = YO(x) in
~
(6.77)
where gS, ¢~ are defined by (5.17), (5.18) and SS by (3.79). Under the present assumptions, (6.77) has been considered in Section 4.3 250
(Theorem 4.7). Let (ys'u s ) E W~"(Q) (6.76), (6.77). We have LEr~t~A
x
Xq be an optimal pair for problem
For s -+ 0
6.3 Us
(6.78)
-+ u* strong!y in Xq
y -+ y* weakly in W2,1(Q) and strongly in C(Q)
(6.79)
SS(ys) -+ fO
(6.80)
s
q
+
6Y*-Yt weakly in Lq(Q).
Proof The proof is essentially the same as that of Lemma 5.2 but we outline it for the reader's convenience. For all s > 0, we have s
s '\.,
H (y ,u ) < H (y ,u*) s s s '\.,
'\.,
where y is the solution to (6.77) with u = u*. By Theorem 4.7, Ys -+ y* 2 weakly ~n W "(Q) and therefore strongly in C(Q). This yields q
lim Hs (y'\., ,u*) = JT0 g(t,y*(t))dt s-+O S
+-
¢(u*)
and therefore lim sup HS(ys'u s) s -+ 0
<J
T
o
g(t,y*(t))dt
+-
¢(u*).
(6.81 )
On the other hand, on a subsequence we have
and by Theorem 4.7 Ys -+ Y1 weakly in W~,l(Q) and strongly in C(Q), where y, is the solution to (6.66) with u = u, . This yields rT
lim inf HS(Ys,ud >- J g(t,y,(t))dt + ¢(u , ) +
Relations (6.79) and (6.80) follows as in 251
Theorem 4.7. Now, returning to the proof of Theorem 6.5, let Ps E H2,1(Q) n L2(0,T;H6(~)) be the solution to the boundary value problem
p s
= 0 in
(6.82)
L
Now since us is an optimal control of problem (6.76) we have T
Io
s
(17g (t,y (t)),w (t))dt s s
+
¢L(U
s
,v)
+.
where Ws E W~,l(Q) is the solution to the boundary value problem (ws)t - 6w s
+-
s's (Ys)w s = ° in
ws (x,O) = 0, x E ~ Ws = v in L1, Ws =
Q
° in L2,
<.,'> is the pairing between Xq and X*q and F~X q + X*q is the duality mapping of X (¢I is as usually the directional derivative of ¢). q Taking the scalar product of (6.82) by Ws and integrating by parts we find, after some manipulation involving Greenls formula, that ap - Ll v a~ dodt..;;
J
¢I(US'V)
+-
Hence (6.83) Now multiplying (6.82) by Ps and then sgn ps we get the estimate IPs(t)
252
12
+
Io
T IIPs(t)
II 2 1
H 0 (~)
dt
+
J'Is s (Ys)psi Q
dxdt..;; C.
(6.84)
Then arguing as in the proof of Theorem 6.3 w,e infer that there exi.sts a function p E BV([O,TJ~H-s(n)), s > N/2, such that on a subsequence s -+ 0, Ps
p strongly in L2(Q), weakly i.n L2(0,T;.H6(n))
-+
and weak star in L (0,T;..L 2(n)). ps(t)
(6.85)
p(t) pointwise in H-s(n) on [O,TJ.
-+
Moreover, there exists ~ E (Loo(Q))* such that on a generalized subsequence, again denoted s, ~s(y)p s s
Thus, going Pt
-+
weak star in (Loo(Q))*.
I.
...
(6.86)
to the limit in (6.82), we see that +
6p -
~
E dg(t,y) in V'(Q)
(6.87)
and p(T)
+
d¢O(y*(T)) 3 0 in
n.
(6.88)
In other words, J P¢t dxdt Q
~(¢)
+
J Vp·V¢ dxdt Q
+
JQ s¢ dxdt - In p(x,T)¢(x,T)dx = 0
+
(6.87) ,
s E dg(t,y*) a.e. in Q~ s E L2(Q) 2
1
2-1
00
for all ¢ E L (O,T;,HO(n)) n L (Q) such that ¢t E L (O,T~H (n)) and ¢(·,O) =0 on n. Now let X E w2• 1(Q) be the solution to the boundary value problem q
X = a in
~1'
X = 0 in
(6.89)
~2
where a is arbitrary in X and XO E w2-(2/q)(n) is such that q
q
xO = a(·,O) in 253
f1 and XO = 0 in f 2• According to the trace theorem (see for instance [49J, O p. 87) we may choose X in such a way that Ilxoll 2-(2!q) I~
q
( n)
'" Clla(',O)1I 2-(3Jq) '" C11lallq . wq ( r 1)
Hence
II X II
2 1
W ' (Q) q
'" CII a
II q
(6.90)
va EX. q
Now we multiply (6.82) by X and integrate on Q. substituting (6.89), we get dp S a dV do dtl
and by (6.90) we infer that dp I ~ a~dodtl",Cllall ov q
J1
Using Green's formula and
Va E X
q
where C is independent of s. This implies that the set {dPS!dV)} is bounded in X*. Then letting S tend to zero in (6.83) we conclude that q
-~ dV E d¢(U*) in ~ 1 where dp!dV E X* is defined as q
J~1
¢
~~ = fQ +
IQ
p¢t dxdt +
J Vp.V¢ dxdt Q
¢dg(t,y*)dxdt -
In
+
~(¢)
p(x,T)¢(x,T)dx
for all ¢ E W~,1(Q) such that ¢(x,O) = 0 for x E nand ¢ = 0 in ~2' Now by the same reasoning as in the proofs of Theorems 5.2, 5.3 it follows by (6.79), (6.80) and (6.85) that 254
PESE(YE)
~ p(fO-Yt
+
6Y*) =
° strongly in L1(Q)
(6.91)
°
1 (6.92) p SE(y)y ~ strongly in L (Q). E E E 2 "(Q) is compactly embedded in C(O) it follows by Lemma 6.3 ((6.79)) Since W q that, selecting a subsequence if necessary, we have y
E
~
y* strongly in C(O)
and by (6.86), (6.92) ~y*
=
° in (Loo(Q))* or (C(O))*.
(6.92)
I
Together with (6.87) and (6.91) this implies (6.72) and (6.74), thereby completing the proof of Theorem 6.5. To give a more specific example, we return to the control problem of oxygen consumption in an absorbing tissue considered in Section 6.3. This time the control function u is the value of the oxygen concentration on the boundary r. Thus we are led to the following optimal control problem: Minimize
(6.93) on all (y,u) subject to the state system
Yt - 6y> -1, y> Yt - 6y = -1
°a.e.
a.e. in [y
y(x,O) = YO(x), x E
~;
in Q >
(6.94)
0]
Y = u in E
and to the control constraints u E Un' where
Uo = {u E Xq ; u(o,O) = YO(o) for Here
ex
/I •
is a positive constant and
° E r,
=
f ~llul12q 1.
+
00
ifuEU 0
° a.e.
in E}.
IIq ;s the norm of X •
This ;s a problem of the form (6.69) where g ¢(u)
u>
q
= 0, ~o(Y) = i ly_yOI~
and (6.95)
otherw; se. 255
Then the subdifferential d~ ( u)
a~
can be written as
:: aFu +- N( u ) VuE U0 :: D( ~) •
where F:X q ~ X*q is the duality mapping of Xq and N(u) normals to Uo at u, i.e. N(u) :: {n E
X~;.
c
X*q is the cone of
(6.96)
We note that X c C(r) for q > (N+-2)/2. Then EU :: {(o,t) ~ r~ u(o,t) > O} is open, and taki~g in (6.96) v :: U ± ¢p where ¢ E C~(Eu) and pER is sufficU iently small we conclude that if ~ E N(u) then ~ = 0 in E (in the sense of measures). Taking v :: U +- ¢ where ~ E C~(r) is positive we see that ~ < 0 in r. Thus by v.irtue of Theorem 6.5 the necessary conditions of optimality in problem (6.93), (6.94) are Pt
6p :: 0 in [y*
+
>
OJ, p :: 0 in [y*
OJ
p(x,T) :: yO(x) - y*(x,T) in n, whilst the optimal control u* is given by
(~
+
a Fu*)u* :: 0, u* >- 0, ~~ +- a Fu* >- 0 in
r.
Equiv.alently u* ::-a-1F-lc~~) in [~~
u* :: 0 i.n [~~
< 0],
>
OJ.
Theorem 6.4 has been established in [7J and [11J in a slightly different form.
REMARK 6.5 equation
By (6.87)' and (6.92)' we see that the dual arc p satisfies the
JQ P¢t
dxdt +-
2
1
r
JQ
'Vp 'V¢ dxdt co
+
r ~¢
JQ
dxdt ::
r p(x,T)~(x,T)dx
)n
2-1
for all ~ E L (O,T~HO(n)) n L (Q) such that ¢t E L (O,T~H (n)), ¢(·,O) :: 0 and ¢ :: in [y* > OJ. Under this form the optimality equations (6.71) to (6.74) have been obtained by Moreno and Saguez [63J (see also [70J, [79J. [80J).
°
256
§6.4 Boundary control of mov.ing surfaces Given the free boundary problems (6.37) and (6.66), denote by Ey the incidence set {(x,t) E Q, y(x,t) = o} and by E (t) the set {x E ~, y(x,t) = OJ. Here y we shall study two problems related to the control of Ey and Ey (t) (see Section ~.5 for the steady case). Problem 1 Given a measurable subset E c Q, find Problem 2 Given a measurable subset ~o = ~o. Here Uo is a bounded, control space U. that Ey(T)
U E
Uo such that Ey = E.
E Uo in a such a way and closed subset of the
c ~, find U con~ex
The least-squares approach to problem 1. leads to an optimal control problem with state equation (6.37) or (6.66) and cost functional
i fQ Ix Ey (x,t)
- XE(x,t) 12 dxdt
+-
(6.97)
l/J(u).
Here XE ' XE are the characteristic functions of Ey ' E and l/J:U y indicator function of UO' i.e. l/J( u) = 1.f
R is the
if u E U
0
+
+
o
co
otherwise.
As seen in Section 6.2, in the case of problem (6.37) the space U might be {u E W,,2([O,T]~L2(r,)), u(O) = O} or U = L2(I,) and Uo = (u E L2(I,), u > 0 a.e. in ~,}. In the case of problem (6.66), U = Xq and Uo is a subset of (6.70). Proceeding as in Section 3.5, we shall approximate the cost functional (6.97) by the following: 1 fQ 1 y+A A - xEI 2 dxdt Z
+.
(6.98)
l/J(u)
which is of type (6.69) where
J
g(t,y) = 21 ~ 1y{x) A+ A - xE(x,t) 12 dx, y
E
L2(~).
A A As seen earlier, this control problem has at least one solution (y ,u ) E
257
L2(Q) xU. PROPOSITION 6.1
([77J) There is a sequence An
An u
+
u* weakly in U
An y
+
2 y* strongly in L (Q)
+
0 such that
where (u*,y*) is an optimal pair in problem (6.97).
Proof The proof is essentially the same as that of Proposition 3.3. we outline it for the reader's convenience. We have
I~ -
I
XE/
2
dxdt <
Q y +-A
I I____
A - XE/
2
(6.99)
dxdt
Q y +A
for all A > 0 and (y,u) E L2(Q) x Uo satisfying (6.37) or (6.66). there exists u* E Uo and An + 0 such that
Hence
A
u n
+
u* weakly in U.
Then by Proposition 4.5 or Theorems 4.6, 4.7, as the case might be, y
An
+
2 y* strongly in L (Q).
Thus selecti.ng a subsequence, if necessary, we hav..e A (y
An +.
n
A) n
-1
+
XE
y*
a.p.. in Q
and by the Lebesgue dominated conv.ergence theorem . L2(II) s t rang 1y ln ~. y* Together with (6.99), this yields An( yAn + An)-1
lim inf A n
and we see
+
0
th~t
+
XE
JQ /A n (yAn
+
A )-1 - XE/ 2 dxdt < n
JQIXE y*
(y*,u*) is an optimal pair.
For problem 2 we consider the optimal control problem:
258
However,
- xEI2
dxdt
Minimize
frt IXEy (T)
Xrt (x) 0
(x) -
2 dx
I
(6.100)
2 over all (y,u) E L (Q) x U subject to (6.31) or (6.66).
o
We approximate the cost functional (6.100) by
J~ !y ( x , ~) +\ 6
_.
XrtO (x) 12 dx
+ 1jJ ( U )
and denote by (YA'u A) a corresponding optimal pair. PROPOSITION 6.2 On a subsequence An u\
+
u weakly in U
+
Y strongly in L2(Q)
+
(
6 • 101)
We have:
0,
n
YA n
where (u,y) is an optimal pair of problem. (6.100).
The proof is identical with that of Proposition 6.1. Now we give an application of the previous theory to the control of the melting front i.n the one-phase Stefan problem (4.68) to (4.70), controlled by the temperature e 1 of the heating medi.um, subject to the constraints (6.102)
le 1 (o,t)! '" 1 a,e. (o,t) E 2: 1 , Consider the following controllability problem:
Find the temperature e subject to constraints (6.102) such that rt-(T) = rtO 1 where ~o is a given subset of rt and rt-(t) = {x E rt, e(x,t) = O} is the solid phase at moment t E [O,TJ.
As seen in Section (4.3), through the Baiocchi transformation y(x,t) = rt
e(x,s)ds,
J j(,( x) O
the one-phase Stefan problem reduces to (6.37)' where fO = - A , Yo = 0 and u(o,t) = f~ e1(o,s)ds for (o,t) E 2:" Thus the controllability problem reduces to problem 1, and according to Proposition 6.2 we may confine ourselves 259
to the optimal control problem with cost functional (6.101) and state equao 1 2 2 tion (6.37) where YO 0, fO ::: -A , U = {u E W ' ([O,T];L (f 1»,U(o,0) = 0 for a E f 1} and
o = {u
U
E U,
lu(o,t)1 . -;; 1 a.e. (o,t) E L 1} •
If (YA'U ) E (W"oo([0,T]~L2(~» n Wl,2([0,T]~V» x U is an optimal pair of A this problem, then by virtue of Theorem 6.4 there exists p E Loo(0,T;L2(~) n 2 L (0,T;H 1(r.» n BV.([O,T]~(V. n HS(~)') which satisfies the system Pt +- lip P
=
~+
d'V
=
0 in [YA
0]
>
0]
in [Y A
0 a.p
dP _ 0 in L 1 ' P ::: 0 in L2' d'V -
p(x,T) =
2(
A
(A+-YA(X,T»
A
A+YA(X,T)
while by (6.65) the optimal control uA is uA(o,t)
o in
(6.103) L3
-x~(X»,XE~, 0 gi~en
by
sgn fT p(o,s)ds a.e. (o,t) E L . t
1
REMARK 6.6 The control problems considered in this section are related to the following problem: For a given surface S = {(x,t) E Q, t ::: ~o(x)} find a boundary function u (subject to some magnitude constraints) such that the free boundary
d{(X,t) E Q, y(x,t) ::: O} of problem (6.66) coincides with S. This is a ill-posed problem, and in the special case of the one-dimensional inverse Stefan problem several numerical algorithms have been proposed in [43]. §6.5
The control of machining processes
We shall study here the following optimal control problem: Minimize T
fo 260
g(t,y(t»dt
+
~(v)
(6.104)
2
2
1
12
on aU y E L (O,T;.H (rt») n C([O,T]; H (rt))
and VE W ' ([O,TLR) subject
to
a(y(t),y(t)-z)
~
frtf(X,t)(y(x,t)-Z(X»dX VZ E K(t), t E [0,TJ(6.105)
where K(t)
= {y E H1(rt);. y> 0 in rt, y = ~(t) in
a(y,z) =
r},
f Vy.Vz dx Vy,z E H1(rt), rl
~:w~,2([0,TJ;'R)
~(v) = {
o +.
+
i.f 0 co
~ is gi~en by ~ Vi ~ P
a.e. in JO,T[
otherwi.se.
CO f E W1,2 C[0,TJ;.L Crt)) and g~[O,TJ x L2(rt) + R+ satisfiES hypothesis (vi). Here W~,2([0,T];'R) is the space {v E W1 ,2([0,T];.R) v(O) = OJ and Q c R3.
As seen in Section 4.4, the variational inequali.ty (6.104) models the electrochemical machining process controlled through the potential difference across the electrodes u(t) = v'et). According to Theorem 4.8, equation (6.105), which can be rewritten as -6y(x,t) > f(x,t), y(x,t) > (6y(x,t) y(x,t)
+.
f(x,t)y(x,t)
a a.e.
in Q = rt
x
JO,T[
= 0 a.e. (x,t) E Q
(6.105)
= v.(t) for x E r, t E [O,TJ,
has a unique solution y E L2(0,T;H 2(rt» n C([O,T];fI'(Q», and the ~ap v + y is compact from W1 ,2([0,TJ;'R) to C([O,TJ;H'(rt»). This implies by a standard method that problem (6.104) admits at least one optimal pair. THEOREM 6.6 Let (y*,v*) be any optimaZ paip in ppobZem (6.104). co 1 2 pEL (O,T;.HO(rt») and E., E L (Q) such that
Then thepe
exist
co 3 6p E (Lco(Q»*, ~~ E L (0,T;.H- / 2(r») (6p - E.,)y* = 0 in Q
(6. '06) 261
I
(6P)a ~(x,t)
==
~
a.e. in [(x,t) E Q, y*(x,t)
>
OJ
(6.107)
E 8g(t,y*)(x) a.e. (x,t) E Q
(6.108) (6.109)
~*I(t)
= 0 if
net)
< o~
= p if
V*I(t)
and 0 < ~*I(t) < p if net)
==
net)
>
0
(6.110)
0,
where n(t) " - ( ds
Ir ~~ vt do
(6.111)
E [O,T].
3 In (6.111) we have denoted by Ir (ap/av)do the value of (ap/av) E H- / 2(r) at 1 E H3/2 (r). In (6.106) we have denoted by (6p-~)y* E (Cl(D»*
«6p-~)y*)(¢)= -J~~
y*¢dx -
J~
Vp.V(y*¢)dx +
Ir y*¢ ~~ do
for all ¢ E C1(~). Now we pause briefly to illustrate Theorem 6,6 on the following model problem: Given a measurable subset E c Q find the potential difference across the electrodes u(t), 0 < t < T subject to the constraints
o<
u(t) <
p , t
E [O,TJ
to ensure on the time interval [O,TJ a minimum mean deviation of the shape
of anode O-(t)
E(t)
==
{x E {x E
~;.
~~
y(x,t) = O} from
(x,t) E EL
As seen in Section 6.4, the least-squares approach to this problem leads us to a optimal control problem of the form (6.104), (6.105) where
The function f is defined by (see Section 4.4)
262
f(x) = {
o The optimality system (6.107) to (6.111) becomes (,~W)a =
p(x,t)
A 2 (XE - _A_) in O+(t) = {x E D; y*(x,t) (A+Y*) A+Y*
= 0 in O-(t) = {x
o u*(t) = {
p
o<
u* <
p
E
~;
y*(x,t)
>
O}
= O}
if net) < 0 if net) > 0 if net) = 0,
(6.112)
where n is defined by (6.111). Proof of Theorem 6.6
As usualy, we start with the approximating problem:
Minimize
fTo g
S(
t , Y( t ) ) dt + 1j! ( v) + 1. "v *- V 112 2 W~,2([O.T];R)
(6.113 )
on all (y,v) subject to
y = vet) in r
where
sS
(6.114 )
is defined by(3.79).
Let (ys'v ) E (L2(0,T;H2(~)) n C([O,T];H1(~))) x U,,2([Q,T];R) be an s optimal pair. Using Theorem 4.8 and its proof it follows as in previous cases that Vt:
+
v* strongly in W1,2([0,T];R)
(6.115 )
Ys
+
y* strongly in C([O,T];H2(~))
(6.116 )
s On the other hand, mUltiplying (6.114), where y = Ys' by IS (ys)IQ-2 Ss(ys) and taking into account the fact that Ss(ys) = 0 in r (see the proof of Theorem 4.7) we get 263
I/Ss(y )11 q ..;; C s L (Q) where C is independent of sand q> 2. Hence {Ss(y )} is bounded in Loo(Q). s For every t E [O,T] and s > 0 there is a unique ps(t) E H~(n) n H2(n) such that ·s s 6p s (t)-p s (t)S (y s (t)) = Vg (t,y s (t)) in n.
00 1 2 Obviously Ps E L (O,T~HO(n) n H (n)). 6.4 we see that
I: Ir v(t)
a:~
(o,t)da dt
+
( 6.117)
Now arguing as in the proof of Theorem
W'(VE'V)
+
I: (v~
- v*')v'dt> a
Vv. E W6' 2( [ 0 , T] ;.R)
( 6. 11 8 )
where ~' is the directional derivative of ~. Now multiplying (6.117) by ps and sgn ps , respectively, we find by a standard method that lip 1100 1 + J Ips(t) II n s L (O,T~HO(n))
6s (y
s
(t)) I dx ..;; C, t E [O,T].
Hence selecting a subsequence (generalized, if necessary) we have (6.119) psSs(ys)
+
w weak star in (Loo(Q))*
-w E L2(Q) and
where 6p
6p - W E dg(t,y*) a.e. in Q.
(6.120)
Now by (6.117) it follows via Green's formula that
Jr
¢ dPs do dV
= J ¢Vgs(t,y )dX+J V¢·Vp dx n
s
D
This yields
Jr 264
¢d:Sdo..;;CII¢111 v
H (n)
2 V¢EH (n)
s
+
J p 6s (y )¢dx n s
s
and therefore by the trace theorem {ap /dV} is bounded in Lcc(0,T~H-3/2(r)). s Thus letting s tend to z.ero in (6.118) we get
-I:
«t)dt
Ir ~~
do"
~'("*,v)
Vv E W6,2(],T];R).
On the other hand, by (6.106) it is readily seen that X E (W~,2([0,T]~R)* of the form X(w) =
I: ~(t)w'(t)dt
d~(V*)
(6.121)
consists of
Vw E W6,2([O,T];R)
where n E L2(0,T;.R) and n = 0 in [0 < V*I(t) < pJ, n>- 0 in [V*I(t) pJ, n '" 0 in [V*I(t) = OJ. Together with (6.121) the latter yields (6.110) and (6.111). Now by inequality (5.65) we conclude as in the ~receding cases that psSs(ys)
+
0 a.e. in Q.
(6.122)
Since {p } is weakly compact in L2(Q) and {Ss(y )} is bounded in Lcc(Q) we infer th~t {p s SS(y s )} is weakly compact in L2(Q). Thus selecting a further subsequence we may assume that
and again by (6.122) we have (6.123) Next by Green's
for~ula,
fQcjl pS(f + 6y S)dxdt = JQcjlp Sf
dxdt - J Vy ·V(cjlp )dxdt Q E S
for all cjlEC 1(Q) and letting S
+
0, by (6.115), (6.119), (6.123) we conclude that
JQcjl p f dxdt -
fQ
1 Vy*·V(cjlp)dxdt = 0 vcjl E C (Q).
Hence
JQcjl p(f + 6y*)dxdt = 0
vcjl E C1 (Q)
265
and (6.109) follows. Now by (5.64) we jnfer that
which implies (6.106), (6.107) by the same reasoning as in the proof of Theorem 5.2. This completes the proof.
REMARK 6.7 Consider the optimal control problem with state system (6.105) and pay-off (6.104) where ~:w~,2([0,TJ~R) ~ ~ is defined as above and ¢0:L 2(n) ~ R+ is locally Lipschitz. If (y*,v*) is an optimal pair then there exist q E H~(n) and 6 E L2(n) such that
(6q)a q
=0
=6
in [x E
in [x E
n;
n;
y*[x,T)
>
OJ
y*(x,T) = OJ
6 E d¢O(y*(T)) P if
v* I (t)
Ir ~ do
oif Ir ~ do JO,P[ if
<
0
>.0
Ir ~J do
= O.
The proof is identical with that of Theorem 6.6.
266
I
7 The time-optimal control problem
This chapter is concerned with the ti~e-optimal control problem for certain control systems governed by parabolic variational i.nequalities. Loosely speaking, this is the problem of steering the initial state of the system to the origin in the minimum time and with control subject to a magnitude constraint. As in the previous chapters, the emphasis is put on the derivation of the maximum principle. The treatment is adapted from the author's work [14J. §7.1
The time-optimal control problem for nonlinear evolution equations
Consider the control process governed by the nonlinear Cauchy problem Y t) + a<jl (y ( t)) I (
:3
u ( t)
a. e. t
>
0, (7.1)
y(O) = Yo in a real Hilbert space H with the norm 1'1 and scalar product (.,.). Here a<jl:H ~ H i.s the subdifferential of an l.s.c. convex function <jl;H + ~ such that 0 E a<jl(O) and yO E ~), u E L~oc(R+;H). Then, as seen in Theorem 1.13, under these conditions (7.1) has a unique solution y = y(t,yo'u) E C([O,oo[ ;R) n w~~~ (JO,+oo[ ;H). Denote by U the class of control functions u, U = {u E Loo(R+;H);lu(t) 1 < r a.e. t
>
O}
where r is a positive constant. A control u E U is called admissibLe if it steers Yo to the origin at some time T (if any), i.e. y(T,yO'u) = O. Before we can proceed any further, we must study the existence of admissible controls. LEMMA 7.1
Assume that yo E TI\¢).
Then there exists at Least one admissible
control.
267
Proof The feedback control law u(t)
= -r sgn y(t), t> a
steers YO to O. Indeed, by virtue of Theorem 1.13 the closed loop system y'(t) + y(O)
a~(y(t))
+ r sgn y(t) 3
a a.e.
t
>
0
= Yo
(7.2)
has a unique solution y E W~;~(JO,+OO[ ;..H) n C(R+;.H) because the operator y -+ a~(y) + r sgn y is maximal monotone. I.n poi.nt of fact it is just the subdifferential a~ of the function ~(y) = ~(y) +-
rlyl, y
E
H.
Now multiply (7.2) by y and use the monotonicity of Iy(t) I' Iy(t) I + rly(t) I -< a a.e. t Let T
lYol/r.
>
a~
to get
O.
If y(t) f 0 for t E [O,T], then this relation yields
Iy(t) 1-< IYol- rt. Hence y(t) = 0 for t > T, thereby completing the proof. The smallest time t for which y(t,yo'u) = 0 i.s called the transition time of the control u. The infimum T(yO) of the transition times of all admissible controls u E U is called the optimal time. In other words,
A control u E U for which y(T(yO)' yO'u) = 0 (i.f any) is called the timeIn this case the pair (y(·,yo,u),u) is called the time-optimal pair. optimal control for system (7.1) with the control constraints u E U.
PROPOSITION 7.1 Undpr thp hypotheses of Lemma 7.1, assume that for every A E R the level set {x E H; ~(x) -< \} is compact. Then there exists at least one time-optimal control for system (7.1).
268
Proof By Lemma 7.1, T(yO) < + 00. This means that there exists a sequence 1 2 {Tn} ~ T(yO) and {un} C U such that y(Tn,yO'u n) = O. Denote by Yn E Wl~c (]O,+oo[ ;H) the solutions to the Cauchy problem
---
(7.3) Yn(O) = YO' We take un = 0 and yn = 0 on [T n , + ooJ and observe that the pair (y n ,u n) satisfies (7.3) on R+. Multiplying (7.3) by Yn and then t y~, and integrating on [O,t], yields via Gronwall IS lemma (see the proof of Theorem 1.13)
Let T > 0 be such that Tn > T for all n > NO' Then the above estimates imply that {Yn} is bounded in C([O,T];H) n W1 ,2(]O,T];H) and for every t E [O,T], {Yn(t)} remain in a compact subset of H. Then by the Arzela-Ascoli theorem we infer that {Yn} is compact in C(]O,T];H) n L2(0,T;H). Thus without loss of generality we may assume that y (t)
y(t) uniformly on every C([o,T];H) and strongly
+
n
in L2(0,T;H) y~ +
2
yl
a~(y n )
weakly in every L ([o,T];H) +
~ = u-yl weakly in ev.ery L2(o,T;H).
As seen earlier, this implies that yl(t) +
a~(y(t)) 3
u(t)
a.e. t
E
]O,T[,
y(O) = yo'
269
where u E U is the weak star limit in Loo(O,T~H) of some subsequence of {u }. n Since Yn(T n ) = a we infer that y(T(yO)) = 0 and therefore u is a time-optimal control. In the case where A = d¢ is a linear operator (or more generally if A is the infinitesimal generator of a Co-semigroup in a Banach space), every timeoptimal control is a bang-bang control (Fattorini [35J) and satisfies a variant of the maximum prin~iple ([36]). More precisely, it has been proved in [36J that if A is the generator of an analytic semigroup, Y1 E D(A) and u* is a time-optimal control for the linear problem YI = Ay + U;
Iu( t) I -<
r, t ;;.. 0,
then for every t E [O,T*[ (T* is the optimal time) there exists pt, a solution to the adjoint equation pi + A*p = 0 on [O,t], such that u*(s) = r sgn pt(s) for s E [O,tJ. Other variants of the maximum principle, together with related controllability problems for linear evolution equations, can be found in the works [4J, [25], [31], [37J, [42], [47], [51J. However, little is known about the validity of maximum principle for nonlinear evolution equations of the form (7.1). We shall see below that the bang-bang principZe and a variant of the maximum principle remain va1id in the general case considered here. §7.2
The maximum principle
Consider the control process governed by the nonlinear parabolic equation Yt(x,t)
+
AOy(x,t)
+
S(y(x,t))
y(x,O) = yO(x) for x
E ~
y(x,t) = 0 for (x,t)
E
r
x
3
1 -
wh ere a.. E C lJ
270
N L:
i,j=1
(~),
a.e. (x,t)
E
~ x R+
R*.
(7.4)
Here S is a maximal monotone graph in R linear differential operator AOY = -
u(x,t)
x R
such that 0 E S(O) and AO is the
( a .. ( x) y ) + a 0 ( x) y , lJ xi Xj 00
a0 E L
(~),
a.. lJ
a .. for all i, j Jl
1 , ... ,N and
N
a 0 >- 0,
L
i ,j =1
a i J" ( x) ~ i ~J"
We shall denote by A:L 2(n) Ay ::: AOY' y E D(A)
;.. w
II ~ II
N a. e. x E Q) ~ ERN.
L2(n) the operator
+
1 2 = HO(n) n H (n)
(7.5)
or equivalently
~
(Ay,z) = a(y,z)
i,j=1
fa""y )[2
lJ
z dx xi. Xj
+
fn aoyzdx
vy,z E H6(n).
Throughout the following we shall denote by H the space L2(n) endowed with the usual scalar product (.,.) and norm' . 1 , 2 Let F:H + H be the operator defined by Fy = Ay
+
B(y)
vy E D(F),
where 1 D(F) = {y E HO(n) n H2 (n);
3 WE
H, w(x) E B(y(x)) a.e. x En}.
As seen earlier (Theorem 4.3), as an immediate consequence of Theorem 1.10 the operator F is maximal monotone in H x H. More precisely, F = 8~ where ~:H + R is given by
~(y) = and 8j = B. yl(t)
i a(y,y)
+
In j(y(x))dx vy E H
In terms of F, (7.4) can be rewritten in the form (7.1), i.e. +
Fy(t)
:3
u(t) a.e. t
> 0
(7.6)
y(O) = YO'
By Proposition 7.1, for Yo E DTF) the time-optimal control problem (P)
inf {T;y(T,yO'u) = 0, lu(t) 12 -< r a.e. t E ]O,T[}
admits at least one time-optimal control. We shall see below that there exist time-optimal controls which are bangbang controls, and in two typical situations these controls satisfy the maximum principle. The first case is where B is locally Lipschitz and satisfies the condition 271
0-< S'(y) -< c(ls(y)1 THEOREM 7.1
+
Iyl
+-
1) a.e. y
E R~
13(0) = 0
Assume that YO E D(~) and 13 satisfies condition (7.7).
1
there exists a time-optimal pair (y*,u*) E (W ,2([0,T*J;H)
H2(~))
x
(7.7)
Then
n L2(0,T*;H6(~) n
Loo(O,T*~H) and p E Loo(O,T*~H) n L2(O,T*,H6(~)) n Cw([O,T*J;H) n
AC([O,T*J;,H-s(~)) satisfying the conditions
(7.8)
1p( t) 12 f: 0 a. e. t E ] 0, T*[;, p' - Ap E L\ Q) and the system
p' - Ap-p3S(y*)
3
0 a.e. in Q = ~
u*(t) = r sgn p(t)
a.e. t
E
JO,T*[.
x
]O,T*[.
rjp(t)!2 - (Fy*(t),p(t)) :: 1 a.e. t Here T* :: T(yO) is the optimal time, s
>
sgn p = P/! pI 2 for p f 0; sgn 0
(7.10) E
JO,T*[.
(7.11)
2N and
=
{w
E
H; Iwl2 -< 1L
In (7.9) pi E L1(0,T*;H-s(~)) is the strong derivative of p:[O,T*] and Ap(t) is the element of H-1(~) defined by (Ap(t),z) = a(p(t),z) for all
(7.9)
+
H-s(~)
1
Z E HO(~).
Thus pl-Ap E L1(0,T*,H-s(~)) +- L2(O,T*~H-t(~)), and so (7.9) simply means that the function t + (p(t),z) is absolutely continuous on [O,T*] for every z E H~(~) and
ft (p ( t) , z)
- a ( p( t) , z ) -, (p ( t) d13 (y* ( t) ) , z ) :: 0 a.e. t E JO,T*[ vz E H~(~).
Hence p satisfies in the sense of distributions the boundary value problem
~ - AOp - p3S(y*) p :: 0 in r
272
x
3
JO,T*[.
0 in Q (7.9)
I
By (7.10), (7.11) we see that u* i.s a bang-bang contY'Ol, i.e. lu*(t) 12 = r a.e. t E JO,T*[. Now we shall consider the case where s(y) = 0 for y
>
0, 6(0) = J-oo,OJ, S(y)
= 0 for y
<
O.
(7.7)'
Then as seen earlier (Section 4.2, Example 4.1), (7.4) is equivalent to the complementarity system AOy(x,t)
= u(x,t) a.e. in [(x,t);y(x,t)
Yt(x,t)
+
Yt(x,t)
= max {u(x,t),O}
(7.12)
a.e. x E 51 (x,t) E r
y(x,t) = 0 We recall that in this case
= {y
OJ
a.e. i.n [(x,t);.y(x,t) = OJ
y(x,O) = yO(x)
D(~)
>
x
R+.
1 E HO(51)~ y(x) > 0 a.e. x E 51}.
THEOREM 7.2 Let yo E D(~). Thqn thepe exists a time-optimal paip (y*,u*) 2 1 (W ,2([0,T*J;.H) () L2(0,T*~H6(51) n H (51))) x Loo(O,T*;H) fop vapiational inr::quaUty
(? .12)
E
and a function
P E Loo(O,T*~H) () L2(0,T*~H6(51)) n BV([O,T*]~H-s(51)), s
> ;
such that p' - Ap E (Loo(Q))*, !P(t)!2 f 0 a.e. t E JO,T*[ and
(p'-Ap)a = 0 a.e. in [(x,t) y*(p'-Ap)
=
E Q~
y*(x,t)
>
0 in V'(Q).
p = 0 a.e. in [(x,t)
E
0].
(7.13) (7.14)
Q; y*(x,t) = 0].
r!p(t)1 2-(Fy*(t),p(t)) =
1~u*(t)
(7.15)
= r sgn p(t) a.e. tE]O,T*[ (7.16)
Equation (7.13) must be interpreted in the following sense: there exists a singular measure v E (Loo(Q))* such that T* r patdxdt + a(p(t),a(t))dt = v(a) + p(x,T*)a(x,T*)dx (7.13)1 )Q 0 Q
f
J
273
1 for ev.ery a E L2(0,T*~H6(~)) n C (Q) such that a(x,O) = 0 and a = 0 in [(x,t) E Q, y*(x,t) = OJ. Equation (7.14) has a similar meaning, namely
J p(Y*~)tdxdt O
J:*
+
= f~
a(p(t).~y*(t))dt p(x,T*)~(x,T*)y*(x,T*)dx
for every ~ E C (Q) such that ~(x,O) 1
§7.3
(7.14)1
= O.
The approximating control process
The approach here is again modelled on the previous developments. The idea is to approximate problem (P) by the following infinite horizon optimal control problem: Minimize
f:
(gs(y(t)) + hs(u(t)))dt
(+ ) 1 2( + ) ovep all u E L2 loc R ~H and y E Wl~c R ~H y' + Ay + SS(y) = u a.e. in ~ y(x,O) Here hS :H
+
= YO(x), x
Rand gS:H
+
x
f'\ II
L2loc (R+'H1(n) ~ O~'
n H2()) ~ subject to
R+ (7.17)
E ~.
R are defined by (7.18)
and (7.19) Here n
E
Coo(R+) is such that nl(y) > 0, 0 , n(y) , 1 for all y
E
R+ and
for y > 2 n(y) = {
o
for 0, y, 1.
LEMMA 7.2 Fop all S sufficiently small, ppoblem (P s ) admits at least one solution (ys'u s )' Proof
274
It is readily seen that there exists at least one admissible pair
(y,u) in problem (P). For instance, gS(yS) ~ hS(u*1) E L1(R+) where yS, u*1 S are defined as in the proof of Lemma 7.3. Hence the value d of infimum in problem (Pc) is finite. Then there exist the sequences {u n }, {y n } such that c-
oo
fo
d,
S
(g (y ) + n
h
(u S
n
))dt , d *
n
-1
•
(7.20)
By (7.18) and (7.20) we see that un remain in a bounded subset of L21oc (R + ;H). Hence on a subsequence we have
and by Lemma 5.1 (see estimate (5.21)), y remain in a bounded subset of 12 + 2 + 1 2 n. Wl~c(R ;H) n L (R ~HO(n) n H (n)). Hence selectlng a subsequence we may infer via the Arzela-Ascoli theorem that for every T > 0, 1 strongly in C([O,T];H) n L2 (O,T~HO(n))
and weakly in L2(0,T~H2(n))
y~
+
y' weakly in L2(0,T;H)
SS(Yn)
+
~
weakly in L2(0,T;H).
This implies that y is the solution to (7.4), where u is the weak star limit 00 + in Ll oc (R ;.H) of some subsequence of {un}' Then by the Fa tou 1emma, lim inf foo gS(y )dt > foo gS(y)dt. n+ oo 0 n 0
f:
On the other hand, since the function u + hs(u)dt is weakly lower semicontinuous on Lioc(R+;H) (because it is convex and continuous) we have lim inf foo h (u )dt> foo h (u)dt n + 00 0 S n O s and by (7.20) we see that
fooo (g
E (y)
= d.
+ h ( u )) d t S
LEMMA 7.3 On a subsequence
S
+
0,
275
LOO(O,T*~H)
(7.21)
u
+
u* weak stap in
Y
+
y* weakly in W1,2([0,T*];.H) n L2(0,T*;..H 2(r2))
S
S
2 and stpongly in C([O,T*];.H) n L (0,T*;.H6(r2))
sS(y€)
t;
+
2 /;.)eakly in L (0,T*;_H)
(7.22) (7.23)
s(x,t) E S(y*(x,t)) a.e. (x,t) E r2
x
JO,T*[
whepe T* is the optimal time and (y*,u*) is an optimal paip in ppoblem (P).
Proof Let (yf,ui) E W1 ,2([0,T*J;.H) x L2 (0,T*;.H) be any optimal pair in problem (P) and let T* be the optimal time (we have already noted that such a pair exists). We extend uf by 0 and yi(t) = 0 for t> T*, and note that Yf is still a solution to (7.6) on R+ for u = uf. :et ~€ be the solution to (7.17) where u = uf. Since hs(ui) = 0 a.e. in R , we have oo roo 'V (7.24) o (gs(ys) + h€(us))dt , )0 gS(Ys)dt,
f
whereas by Lemma 5.1, (7.25)
'" and u = u1' by '"Ys and taking into account Now multiplying (7.17), where y = Ys the coerciv.ity property ( Ay ,y) >
w
II y II 21
v Y E H6(r2 )
n H2( r2) ,
HO ( r2) one obtains the estimate
because uf = 0 on [T*,+oo[. Together with (7.25), this yields 1 2 ';s(t) 12 ,Cs / vt > T*. Then by (7.25) and the definition of gS it follows that for all sufficiently small s, 276
Together with (7.24), this yields rOO
lim sup €+ 0
J (g€(y) +- h (u ))dt -< T*. €
0
(7.26)
€ €
Since {u } is bounded in L2l (R+-~H) it follows by Lemma 5.1 that there exists 2 € + oc u* E Lloc(R ~H) such that for every T > 0 and on a subsequence E + 0, uE + u* weakly in L2(O,T~H), YE + y* weakly in W1,2([O,TJ;H) n L2(O,T~H2(~))
(7.27)
and strongly in C([O,TJ~H) n L2(O,T~H6(~))
(7.28)
where y* is the solution to (7.6) with u = u*. We shall prove that u* is a time-optimal control. To this end we note that by (7.19) and (7.26) it follows that the Lebesgue measure of the set {t E R+~ly€(t)l~ > 2El/4} is smaller than T*. Hence there exist €n + 0 and tn E [O,2T*J such that /y
En
2 1/4 (t )/2 -< 2E for all n. n n
(7.29)
Selecting further subsequence we may assume that tn + TO. On the other hand, Since {y~ } is bounded in every L2(O,T;H), we have n
1yE ( t) - y E (t) 12 -< CIt - t n /1 / 2 v t n n
E [
0 ,TO] .
n
" = inf{T;y*(T) =OL Then by (7.28) and",(7.29) we infer that y*(T O) = O. Let 'T We will show that T = T*. To this aim, for every E > 0 define the set 1 E€ = {t E [O,TJ;IYE(t) I~ > 2E / 4 }. By (7.26) we see that
where m denotes the Lebesgue measure. for otherwise there would exist 0
>
" On the other hand lim sup m(EE) = 'T,
0 and En
€ +
+
a
'"
0 such that m(EE ) -< T-o. n
other words, there would exist a sequence of measurable subsets An
C
In '"
[O,TJ 277
such that m(A ) > 8 and iy (t)1 22 , 2E 1/4 for t E An' n En n imply that 1/4 1/2 \y*(t)1 2 , ( 2E n) ~
where
+- ~n
for t
E
By (7.28) this would
An' rv
n
+
O.
On the other hand, since y*(t) j 0 for t E [O,T] we have
lim m{t
n+ oo
E
[O,T]~ly*(t)12' (2E~/4)1/2 + ~n} = O. rv
The contradiction arrived at shows that indeed lim sup m(EE) = T and thererv
E +
0
fore T = T*. Hence u* is a time-optimal control. Relations (7.21) to (7.23) follow by (7.27) and by Lemma 5.1. The proof is complete. 2 ++ Note that by (7.18) and (7.26), uE E L (R ~H) + L (R ~H). Then by some manipulation involving (7.17) it follows that y E Loo(R+~H) and therefore 00
E
GE ( y E)
= 2y E E -1/
4
1T'
(
I '2 ~~ ~)
E L00 ( R\H ) .
( 7 • 30)
E
Recalling that the derivative BE of BE is positive we infer that there exists a unique function pE E Loo(R+;H) n L12 oc (R+~H01(~) n H2(~)), which . n W11,2(R+~H) oc satisfies the equation (7.31 )
(7.31)'
Since (UE'YE) is optimal in problem (P E) and hE' gE are Frechet differentiable functions, it follows by a standard method that (see for instance Theorem 5.7)
Equivalently (7.33)
Equations (7.31), (7.32) can be viewed as first-order conditions for optimality in problem (P E). 278
Consider the optimality system (7.17), (7.31), (7.33), i.e. y' + Ay + SE(y ) E r sgn p ESE S P~
·E
S
- APE - S (YS)PE = G (YE)
+
sp
E
a.e. t
>
0
a.e. t
>
0
(7.34)
and notice that (see Lemma 1.2)
Then by (7.34) it follows that
(p (t),Ay (t) + SE(y (t))) + gS(y (t)) E SSE (7.35) Multiplying (scalarly in H) the first equation in (7.34) by Ay S + SE(y S ) and integrating on [O,t] we get
J:
!Ay£(s) + S£(y£(s))
!~dS
<: C(t+!) for all t> O.
(7.36)
2 + On the other hand, we see by (7.18) and (7.32) that PE E L (R ;H). with (7.36) the latter yields
m{t E [O,n]; I(AYE(t)
+
Together
SE(YE(t)), ps(t))1 > n- 1/ 4 } < Cn 3/ 4
for all natural numbers n. Thus for every n the interv.al [n 1/ 6 ,n] contains a subset En such that m(E n) > 1 and (7.37) 2 In as much as gS(ys) E L1(R+-) and Ps E L (R+;H), this implies that the lefthand side of (7.35) converges to zero for a sequence t ~ + Hence n
00.
E rip S(t) 12 + -2 IPS (t) 122 S = (p (t), Ay (t) +SE(y S(t))) +g E(y E(t)) Vt > O. E (7.38) 279
On the other hand, again using system (7.34) we get
~t (p (t),Ay (t) ~ S((y (t))) Ul. ( ( ( ( p • ( t) ,Ay (t ) + SE(y (t)) + ( Y• ( t) ,A p (t ) + (E
(
(E
B( (y E (t)) p( (t))
(Ay (t) + S((y (t)),G((y (t)))+(r sgn p (t)+(p (t),Ap (t) (
( (
E
(
(
+ p (t)({(y (t))) >- 0 (
(
because SE>-
a and
S((y )y >- O. (
Together with (7.37), this implies that
(
(p (t), Ay (t) + s(( y (t))) -< a v t >- a ( (
(
so that (7.38) yields (7.39 ) Let A E ]O,T*[ be arbitrary but fixed.
By (7.30) we see that
Since y + y* in C([O,T*]~H) and y*(t) j 0 for t E [O,T*[, we see that ( GE(y((t)) = a for t E [O,T*-A] and 0 < E < (O(A)' Hence p'(t)-Ap (t)-p (t)S((y (t)) = 0 a.e. t (
(
E
(
E
]O,T*-A['
(7.40)
Taking the scalar product of (7.40) by P (t) and integrating the result on ( [O,T*-A]' by (7.39) we get
jT*-AII p
2 1p (t) 12 + 2w (
a<
t
(t)
E
II 2 1 H
(~)
2-2 dt -< 1p (T* - A) 12 -< (rw)
o
(
(7.41)
(O(A) and 0 -< t -< T*-A' Now we multiply (7.40) by ~(p() where ~:R + R is the smooth approximation (3.66) of the signum function. Integrating the result on Q = ~ x ]O,T*-A[ and using Green's formula we get A for
( <
fQAp( (x,t)S((Y where jO(r) = fr
o 280
E
(x,t)) ~(p (x,t))dxdt -< (
f
~
jO(p (x,O))dx, (
~(s)ds. Letting ~ tend to sgn we get for some constant C
independent of sand \ 's
Ip s (x,t)B (y s (x,t»ldxdt -< (mU2)
1/2
Ip s (0)1 2 -< C, (7.42)
By (7.41) it follows via a selection principle that there exi.sts pELoo(O,T*;H) n L2(0,T*~H~(n» such that on some subsequence En +
°
p + p weak star in Loo(O,T*~H) sn p + p weakly in L2 (0,T*;H 1(n» 0 En 2
APsn + Ap weakly in L (O,T*;H
-1
(7.43) (7.44)
(Q».
(7.45)
¥
Recalling that H~(n) c C(D) for s > we have L1(n) c H-s(n), and by (7.40), (7.41), (7.42) we see that {p'} is uniformly bounded in every L1(0,T*-\; s H-s(n». Since the injection of H into H-s(n) is compact it follows by the vectorial Helly theorem that, on a subsequence, {PSn} converges pointwise on [O,T*[ to the function p E BV[O,T*[;H-s(Q», i.e. Ps (t) + pet) strongly in H-s(n) and weakly in H n
for every t E [O,T*[. This implies as in Section 5.2 (see (5.35»
(7.46) that (7.47)
Let V([O,T*-A],p) be the vari.ation on p on [O,T*-\]. Since the function \ + V([O,T*-\J,p) is bounded (because {p~} is uniformly bounded in L1(0,T*-A~H-s(n» and monotonically increasing, it follows from the inequa1 ity < til < t' < T*,
°
-< V([O,t'];.p)-V([O,t"];p),
IIp(t')-p(t") II -s H
(n)
that lim pet) = p-(T*) exists in the strong topology of H-s(n). We extend t+T* (or redefine as the case might be) p on [O,T*J by p(T*) = p-(T*). The function p so defined is of bounded variation on [O,T*J, and (7.46) extends 281
to all t E [O,T*J. Now by (7.38) we see that for all 0 rip S(t)1 2
<
S
<
SO(A) we have
-2S Ip S(t)1 2 2 =(P S(t),Ay S(t)+_Ss(y S(t)))+-l a.e. tEJO,T*-A[.
+.
Letting S tend to zero and bearing in mind (1.23), (7.46) we get (7.11), i.e.
rlp(t) 12 - (p(t),Fy*(t)) = 1 a.e. t E JO,T*[ In particular, it follows that pet) f 0 a.e. t E JO,T*[. tend to zero in (1.33) we see by (7.21) that
(7.48) Then letting S
u*(t) = r sgn pet) a.e. t E JO,T*[. Hence lu*(t) 12
(7.49)
= r a.e. t E JO,T*[. On the other hand, since by (7.33)
lim lu (t) 12 = r a.e. t E JO,T*[ s-+O S it follows by (7.21) that on a subsequence, again denoted sn' we have Us -+ u* strongly in L2(0,T*~H) and weak star in Loo(O,T*~H) n
USn(t) -+ u*(t) a.e. t E JO,T*[ strongly in H. Let An -+ 0 for n -+ ~n
=
00
(7.50)
and let {~ n } c L1(Q) be defined by
,sn S (Ys )ps in QA ' ~n n n n
=
a in
Q'QA . n
According to estimate (7.42), {~ } is a weak star compact subset of (Loo(Q))*. n Hence there exists ~ E (L (Q))* and a generalized subsequence {~} of {sn } such that 00
~
~
-+ u weak star in (Loo(Q))*.
(7.51)
Then going to the limit in (7.40) we infer that p' - Ap -
~
= O.
(7.52)
This means that p' E V'(Q) admits an extension in (Loo(Q))* + L2(0,T;H- 1(n)) which satisfies (7.52). More precisely, we have 282
Jr patdxdt
+
Q
for all a
E
JT* 0
a(p(t), a(t))dt
+ ~(a)
J
p(x,T*)a(x,T*)dx (7.52)'
J
C'(Q) such that a(x,O) - O.
Summarizing, we have PROPOSITION 7.2 There exists a time-optimal pair (y*,u*) given by (7.21), (7.22) which satisfies, together with ~ E (Loo(Q))* and p E Loo(O,T*;H) n 1 ) n BV([O,T*];H -s (~)) defined by (2.43) to (7.46) and (7.51J, L2 (O,T*;.HOU'I) equations (2.49) and (7..52).
In brief, Proposition 7.1 is tantamount to saying that if 6 is a general maximal monotone graph hav.ing the property that 0 E 6(0), then problem (P) has at least one optimal control u* which satisfies a weak form of the maximum principle, i.e. (7.49), (7.52). in particular, it follows that u* is a bang-bang control.
We know from Theorem 5.7 that the optimal control uE is a feedback optimal control of the form uE(t) E -Ea~E(YE(t)) - r sgn a~E(YE(t)), where, for every Yo E H, cpE(yO) is the optimal v.alue of problem (P E). by virtue of Lemma 7.3, ~E(yO) + T* for E + 0, we may v.iew u
= _Ea~E(y)
Since,
- r sgn acpE(y),
as an approximating feedback optimal control for problem (P). Section 5.8, ~E is the solution to the Bellman equation.
As seen in
E (acp E (y),Ay +6 (y)) +-l r acp E(y) 12 +2E lacp E(y) 122 = gE(y). §7.4
The proof of the maximum principle
Since the proofs of Theorems 7.1,7.2 are essentially the same as those of Theorems 5.' and 5.2 they will be outlined only. Proof of Theorem 7.1
As noted earlier, if 6 satisfies condition (7.7) then (7.53) 283
This yields
jEIp -<
(x,t) IsS(y (x,t)dxdt S
S
C(j
E
J
Ip (x,t)1 ISE(y (x,t))ldxdt +Ip (x,t)y (x,t)ldxdt s S E E E +
JE Ip
(7.54)
(x, t) Idxdt), S
where E is any measurable subset of Q and {y
}, {p } are the-sequences sn sn found in the preceding section. Since {p } and {y } are strongly converS S sn n 2 n 2 gent and {S (y )} is weakly convergent in L (O,T*;H) = L (Q), we see by sn (7.54) that for every n > 0 there exists a(n) independent of n such that ·sn Ip (x,t) s (y (x,t)) Idxdt -< n E En En
J
if m(E) -< a(n). sequence {Ssn(y ~
1
S
Then by the Dunford-Pettis criterion we conclude that the )p } is weakly compact in L1(Q) C L1(0,T*;H-s(~)). Hence S
n
n
E L (Q), p E AC([O,T*];H .Sn S (Ys )ps + n
~
-s
(~))
and (7.51) is strengthened to 1
(7.55 )
weakly in L (Q).
n
On the other hand, by Proposition 7.2 p E L2(0,T*;H6(~)) n Loo(O,T*;H). Since p is bounded from [O,T*] to H and continuous from [O,T*] to H-s(~) we infer that p E Cw([O,T*];H). Since (7.8), (7.10), (7.11) have been established in Proposition 7.2 it remains to prove that ~(x,t) E
p(x,t)as(y*(x,t)) a.e. (x,t) E Q.
By the Egorov theorem, for each n > 0 there is a measurable subset such that {y } are uniformly bounded on E and y (x,t) + y*(x,t) sn n sn ·sn in (x,t) E E for S + O. Hence S (y ) are uniformly bounded on n n sn without loss of generality we may assume that .Sn
S (y
sn
) +g weak star in Loo(E). n
(7.56 ) E c
n
Q
un iformly E , and n
(7.57)
Then by Lemma 3.4 we may conclude that g(x,t) E as(y*(x,t)) a.e. (x,t) EE . n
284
Since {p } is strongly convergent to p in L2(Q) we have by (7.55) and (7.57), S
~(x,t)
= p(x,t)g(x,t) E p(x,t)aS(y*(x,t)) a.e. (x,t) E En
and (7.56) follows.
Thus the proof of Theorem 7.1 is complete.
Proof of Theorem 7.2
In this case we have (see (3.79))
S€(r) = s-1 J1 (r-s 28)p(8)d8 s-2 r
+
sJ1 8p(8)d8 for r E R. 0
This yields (see (5.64), (5.65)) pJSs(y )-y SS(y )) = _ sp c..
s
€
S
S
(rJ
1
8p(8)d8
-2
+
f1 8p(8)d8) 0
+
0,
Y€
€
strongly in L2(0,T*;H).
IYS(x,t) I)
-+
2slps(x,t)
1
(7.58)
a.e. (x,t) E Q,
(7.59)
where
{
o 2
if ys(x,t) < - s . Let {sn} -+ 0 be the sequence found in Section 7.3, i.e. the sequence for which (7.43) to (7.47) hold. It follows by Lemma 7.3 that 2 is bounded in L (Q). Then by (7.42), (7.59) we may select further a subsequence sn such that Ps (x,t)S n
sn
(Ys (x,t)) n
-+
0 a.e. (x,t) E Q.
Now by (7.23) and (7.47) we have En PE B (ys) n
where
~
-+
1
p~
weakly in L (Q)
n
= u* - Yt - Ay*
E
S(y*) a.e. in Q.
As seen earlier in the proof of 285
Theorem 5.2, this implies that Sn
P S (y ) sn sn
-+
p~
= 0 a.e. in Q and therefore
1 0 strongly in L (Q)
(7.60) 0 a.e. (x,t) E Q.
p(x,t)(u*(x,t)-Yt(x,t)-Ay*(x,t))
(7.61 )
In as much as Yt(x,t)-Aoy*(x,t)
0 a.e. in [(x,t);y*(x,t)
OJ
and by (7.49) u*(x,t) = p(x,t)/Ip(t)
12 a.e. (x,t)
E Q,
we conclude that (7.15) holds. Next, by (7.58) and (7.60) we see that .Sn
1
(7.62) Ps ys S (Y sn ) -+ 0 strongly in L (Q). n n According to the Egorov theorem, for every 0 > 0 there exists a measurable subset Ho c Q such that m(~~) ~ 0, {y } are uniformly bounded on H~ and u sn \) for sn -+ 0, Ys (x,t) n
-+
y*(x,t) uniformly for (x,t) E Ho.
Then selecting a generalized subsequence from {s n } we may infer as in the proof of (7.51), (7.52) that
= 0 on every Ho
y*~
where ~ E (Loo(Q))* satisfies (7.52). In other words,
fH
~a(x,t)y*(x,t)~(x,t)dxdt o
+
~s(y*~) = 0
(7.63)
00
for all ~ E L (Q) which vanish outside Ho. The 'singularity' of ~s means that there exists an increasing sequence of measurable subsets Ek c Q such -1 00 that m(Q,E k ) ~ k and Ws 0 on L (E k). By (7.63) it follows that r J
HonEk 286
Way*~dxdt
0
for all ~ E Loo(Q) with support in H8 n Ek.
and letting 8
= 0 a.e. (x,t) E H8 n Ek
~a(x,t)
y*(x,t)
0, k
+
~
y*(x,t)
a
+
This yields
00
we get
(x,t) = 0 a.e. (x,t) E Q
and (7.13) follows. To prove (7.14) we multiply (7.40) by Y2~ and integrate by parts on QA' Using (7.62) we conclude that for 2 + 0 n T*-\ p (y ~)t dxdt + a(p ,y ~)dt Q C n En 0 En En
j
J\
-J
Q
PE (x,T*-\)YE n
n
(x,T*-A)~(x,T*-\)dx
+
0
for ev.ery A > O.
Together with (7.22), (7.44) and (7.46), this yields T*- A Q p(Y*~)tdxdt + j0 a(p,y*~)dt
f
= J p(x,T*-A)y*(x,T*-\)~(x,T*-A)dx Q
and letting A tend to zero we obtain (7.14), thereby completing the proof. REMARK 7.1 If N = 1 then by (7.22) we see that Yc + y* uniformly in Q (i.e. in C(Q)) and by (7.51), (7.62) we infer that ~y* = 0 in Q. Thus in this case (7.13) becomes y*(p' - AOp)
= 0 in Q.
REMARK 7.2 Theorems 7.1, 7.2 and Proposition 7.1 remain valid if the homogeneous Dirichlet condition in (7.4) is replaced by a general linear boundary value condition of the form
where a i > 0 for i = 1,2; a 1
+
a
2 > O.
287
REMARK 7.3 Theorem 7.2 allows, by a minor modification of the proof, a natural extension to more general nonl ineari.ties of the form
where 81 satisfies condition (7.7) and 82 is the graph defined by (7.7)1. The details are left to the reader. §7.5
Various extensions
(1) The argument above could clearly be applied to the time-optimal problem associated with the equation Yt(x,t) + AOy(x,t) = u(x,t) a.e. (x,t) E ~ y(x,O) = YO(x), x E
~
~
x
R+ (7 .64)
r
+ 8(y) 3 0 a.e. i.n
x R+,
where 8 is a maximal monotone graph in R x R. For instance, the thenmostat control problem leads to a problem of this type where 8 is given by (7.7)1 and which corresponds to nonlinear boundary value conditions of Signorini type, i.e. dy
dy _
y >- 0, dV>- 0, Y dV - 0 a.e. i.n
(7.65)
In this case, repeati.ng word by word the proof of Theorem 7.2, we find that there exists an optimal pair (y*,u*) E W1,2([0,T*];.L2(~)) x Loo(O,T*;H) for the time-optimal control problem associated with (7.65) and a function
p E Loo(O,T*;.H) n L2(0,T*;H1(~)) n BV([O,T*];.(Hs(~))I) such that ip(t)1 f 0 2 a.e. t E O,T*[, E (Loo(E))* and
;g
pi - AOp
=0
in Q = ~
x
dY* p av- :: 0 a.e. in E = r
( dp
d)a y*
u*( t)
]O,T*[ x
]O,T*[
= 0 a.e. in E r sgn pet) a.e. t E ]O,T*[.
(7.66) (7.67) (7.68)
(7.69)
As for Theorems 7.1, it allows in this context the following formulation: 288
there exists a time-optimal control pair (y*,u*) E Wl,2([0,T*J~L2(D)) x 00 00 2 1 n L (O,T*;.H (~)) n Cw([O,T*J;H)
L (O,T*;.H) and a function pEL (O,T*;H)
n
AC([O,T*J;(Hs(D))') satisfying the conditions Ip(t)1
2
I 0 a.e. t E JO,T*[, ~~ E Ll(E)
(7.70)
and the equations
p' - AOp = 0 a.e. in Q
(7.71)
~ ~ 0. a. e ,'n ~~ dV - p~B(y*) OJ
(7.72)
u*(t) (2)
= r sgn pet) a.e.
(7.73)
t E JO,T*[.
Time-optinldl corltrol of finite-dimensional v.ariational inequalities.
Consider the time-optimal control problem (P) in the special case of system (5.85), i.e. inf {T;.y(T,yO'u) = 0, i1u(t)
II N -<
(7.74)
r a.e. t E JO,T[}
where y(t,yo'u) is the solution to the complementarity system y
i (t)
y.(t) >- 0, 1
y.(O) 1
= u i ( t)
+ (Ay ( t) ) i
y~(t) + 1
= y.1 , 0'
i
a. e • ; n [ t ;.y i ( t) > 0J
(Ay(t)). >- u.(t) a.e. t E JO,T[ 1
(7.75)
1
= 1, .•• ,N.
Here A is a positive definite matrix of dimensions N x Nand y. 0 >- 0 for 1,
a11 i. THEOREM 7.3 There exists one time-optimal pair (y*,u*) E W1 ,2([0,T*J;R N) x N Loo(O,T*;R ) for problem (P) having the property that th8re exists pEBV([O,T*J; RN), pet) I 0 a.e. t E JO,T*J which satisfies the system pi(t) - (A*p(t));
=0
in [t;yi(t) > 0], i
= 1, ... ,N
Pi ( t )
0 a. e. i n [t; yi ( t) = 0], i = 1, ... , N
ui(t)
r Pi(t)/ IIp(t) II N a.e. t E JO,T[, i
rJ Ip ( t)
=
(7.76) (7.77)
1, ... ,N
II N -
(7.78) (7.79 ) 289
Here (.,.) is the usual Eucl i.dean scalar product in RN and T* i.s the optimal N time. Equations (7.76) should be understood in the sense of distributions in the open subsets [t E JO,T*[;y~(t) > OJ. 1
The proof is essentially the same as that of Theorem 7.2. it for the reader's convenience. We start with the approximating problem~
However, we sketch
Minimize rOO
J (gE(y(t» + hE(u(t»))dt
(7.80)
o
12+N
2(+N
on aZZ y E Wl~c(R ~R ) and u E Lloc R ~R ) subject to
y'
+.
Ay
+.
yE(y) = U a.e. t
>
0
(7.81 )
where yE:RN + RN has been defined in the proof of Theorem 5.5 and gE:RN hE:R N + R are given by (see (7.18, (7.19»
R,
+
hE(u) :: (2E)-'(ll u Ii - r)+)2, u E RN N 2 gE(y) :: 1T( Ily 1/ E- 1/ 4 ), y E RN N
Arguing as in the proofs of Lemmas 7.2, 7.3 it follows that, for every E problem (7.80) has an optimal pair (YE'U E), and for some E + 0, 00
uE + u* weak star in L YE
+
(O,T*~R
>
N
)
y* weakly in W1 ,2([0,T*J;R N) and strongly in C([O,T*J~RN).
Let PE be a dual extremal arc in problem (7.80), i.e. E 'E( PE, - APE - Y YE ) PE -- G ( YE ) a.e. t 00
+
N
> 0
(7.82)
PE E L (R ;.R ). Then we have (7.83)
290
0,
and proceeding as in the proof of Proposition 7.2 we get (see (7.35),
(7.38))~
On the other hand, for every A E ]O,T*[ we have
for all E sufficiently small. Multiplying the last equation by PE,: and sgn p~c.., 1. in turn, we get the estimate (see (5.98))
where C is independent of A and E. Now arguing as in the proof of Theorem 7.2 (see also the proof of Theorem 5.5) we may pass to the limit in (7.82), (7.83) to conclude that there exists a function p satisfying (7.76) to (7.79).
291
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