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0, the classical theory (as, for example, in Agmon, Douglis, and Nirenberg [1]) applies: If every/ e L 2 (H) and . We therefore have Theorem 2.3. 0+. Moreover, uniform asymptotic stability of this system will follow from the results of Chapter 3 and the fact that a>K -> w 0 > 0 as K->0+. 2. The limit of the M-T system as K -> 0+. We shall consider the system (3.2.1)-(3.2.3) and denote its solution by {if/K,
THEOREM 2.3. Assume that FeC 1 ' 1 , F o nr,=0, and that the gains f ^ e C\r,). Then the conditions ue Y, ve Y, Au+ five % imply ue[H\Sl)]3.
Therefore, under the hypotheses of Theorem 2.3, the solution of Theorem 2.1 is classical. What can be said about regularity of D(M) in the case F^C 1 ' 1 ? Let us assume that Fe C1'1 except at a finite number of corners P,, • • • , Pn, each of which is formed by a pair of line segments in F. At each corner one may introduce polar coordinates (r,&) such that, locally, ft is a sector S, = where r is the distance to the corner and where
34
CHAPTER 3
•&t € (0,217) is the angle at Ph measured inside of ft, formed by the line segments which produce the corner. Thus the study of regularity of solutions of (2.19) in a neighborhood of each Pt becomes the study of regularity of (2.19) (written in polar coordinates) in the sector 5,. In the case one has either Dirichlet data on both edges of Sf or Neumann data on both edges. Problems of this sort have been extensively investigated by Grisvard [1] and others. To apply the results to the present problem, we set in (2.18) and note that the w component of u = {
where
and
Introduce
such that
Then w - W satisfies
where Regularity of solutions of (2.22) in a polygonal planar region fl has been studied by Grisvard [2]. In particular, if fl is a sector with interior angle d0 and either Dirichlet data on both edges (w- WeHo(£l)) or Neumann data on both edges (w - We H\fl)) the singular behavior of w - W near r = 0 is of order r"7*0. Since ^0<2tr and WeH2(£l) we may conclude that
Note that In particular, if we have corresponds 11 to the C ' situation.) Next, we set w = 0 in (2.18). Then we see that the {ty,
where
and
In fact, (2.23) is just the equilibrium system
MINDLIN-TIMOSHENKO PLATES
35
of plane elasticity, i.e.,
As above, one may assume that hl = h2 = 0. Grisvard [3] has kindly communicated an analysis of the singular behavior of the solution to (2.23) in a sector (1 with angle 60. In the case of Dirichlet data (2.25) or Neumann data (2.26) (with hl = h2 = 0) on both edges of the sector, the result is that {iff, tp} behaves as rz near r = 0, where Re z> 5 depends on the angle 0 0 e (0, 2 IT). The value of z is determined from a transcendental equation obtained by substituting {$,
36
CHAPTER 3
the set of indices j for which y, is contained in F0, and N denote the complement the anele. measured toward the interior of D in II, 2 • • • , «}. Denote by and when formed b\ We assume of In addition, we define and we assume that We then have the following theorem. THEOREM 2.5. Assume that (2.29) and (2.30) are satisfied. Then there exists * 0 >i such that ue V, ve T, Au + Bve W implies ue[Hs(fl)]3 for s<s0, s^2. Theorem 2.5 is a consequence of the results of Grisvard [3] and the decoupling procedure described in the last section. In fact, the only situation that has to be checked is the regularity of the solutions of (2.22) and (2.23) in a sector with interior angle 00 when homogeneous Dirichlet data is given on one edge and homogeneous Neumann data on the other edge. For the problem (2.22), the results of Grisvard [3] show that the singular behavior of w — W near r = 0 is of order r^72"0, so that w e H s (fl) for s < s0 = 1 + TT/200, s g2. In particular, The analysis of the singular behavior of solutions of (2.23) in a sector with mixed Dirichlet and Neumann conditions on the two edges is considerably more complicated than the case of pure Dirichlet or Neumann data described in the last section. However, in this case also the order z of the singularity r z in solutions of the form{i/>,
where A and a are the Lame coefficients. (These are given in terms of E and IJL, by 2cr = £/(l + /Lt), A =2
MINDLIN-TIMOSHENKO PLATES
for finitely many corners formed by straight line segments in and F, always meet at a strictly convex corner.
37
provided
2.4. Strong asymptotic stability of solutions. 2.4.1. The case T 0 ^ 0. Since the operator -'^T'.stf generates a contraction semigroup on *Fx 3f, strong asymptotic stability of solutions of (2.10) may be investigated by use of the Nagy-Foias-Fogal decomposition theory as in, for example, Benchimol [1]. (Strong asymptotic stability means that exp(-«" 1 ^)U°^0 strongly in TxX as t-*oo, for all U0eYx%. Equivalently, E ( t ) -» 0 as t -> oo for every solution of (2.1), (2.2).) If the resolvent of -c$~lsd is compact, application of this theory amounts to showing that -m^jtf has no spectrum on the imaginary axis. (Since —^~ls^ is dissipative, it has no spectrum in the half-plane Re (A) > 0.) We shall assume that (O, F01^) are such that -<€~lsd does have compact resolvent. Conditions that assure this property were given in the last section. The eigenvalues of —m^st are determined by the spectral equation where i.e.
is imaginary, (2.32) becomes
where °V now stands for the complexification of the real space Y. If u> = 0, clearly u = 0 since F 0 ^ 0. Suppose that
satisfy
38
CHAPTER 3
Equations (2.37) and (2.38) imply that {i^,
However, solutions of (2.36) are (real) analytic in (1 (since the system is elliptic with real constant coefficients). This fact, together with (2.39), implies that in Therefore, we have the following result. THEOREM 2.6. Assume that F is Lipschitz, that T0 ^ 0, and that that the gain matrix F satisfies (1.9) and is positive definite on some nonempty, open subset of r\. Then every finite energy solution of (1.1)-(1.3), (1.8) satisfies
one does not have energy decay to zero 2.4.2. The case of every finite energy solution of (1.1)-(1.3), (1.8), regardless of how the gains are chosen in (1.8). Indeed, define where a, j8, and y are nonzero constants. It is easy to check that [ifi1, tp\ w1} is a solution of (2.36), (2.37) corresponding to /j, = 0 (and every such solution is of this form). Let {^°,
and let us set
Then we readily verify that {(/*,
(dK^ is the subspace of 3V orthogonal to the eigenfunction {i/r1, ipl, w1} corresponding to the eigenvalue A = 0.) The key point is that the bilinear form
MINDLIN-TIMOSHENKO PLATES
39
a(«/>,
The key point is the following: If is positive definite on some subset of of positive measure, then This is strictly coercive on is a consequence of the following lemma LEMMA 2.3. Let y be a subset of F having positive measure. Then (a) there such that for all exists
(b) For every all
such that for all
there exists
and for
Proof. If (2.42) is false, there is a sequence
such that
and
Upon extraction of a subsequence (still denoted by weakly in
we have
and strongly in
From (2.44), (2.45), and Korn's Lemma (2.6), we conclude that and therefore,
40
CHAPTER 3
Also, from (2.46) and (2.44) we obtain
But
if and only if
and then the right-hand side of (2.48) gives Then which contradicts (2.46). Equation (2.43) is proved similarly. With the feedback law (2.40), solutions of (1.1)-(1.3), (2.20) are defined by
With initial data the problem (2.49), (2.50) has a unique solution. We may prove, exactly as in the last section, that if F is positive definite on some nonempty, open subset of T, then the solution of (2.43), (2.44) satisfies
be THEOREM 2.7. Assume that is Lipschitz and that a solution of the closed loop system (1.1)-(1.3), (2.40) with initial data in Assume that F and F are positive semidefinite on F; that F is positive definite on a nonempty open subset of F; and that F is positive definite on a subset of F of positive measure. Then Remark 2.6. The modification (2.40) of the feedback law to handle the case was suggested by Zuazua [2] in the context of stabilization of the wave equation. A similar device was also employed in Lagnese and Lions [1] to handle noncoercive situations in the context of exact boundary controllability of M-T plates. Equation (2.40) may also be used as a stabilizing feedback on when In this case (2.41) is replaced by
and it is enough to assume that
is positive semidefinite on
3. Uniform asymptotic stability of solutions. 3.1. Geometric assumptions. We wish to establish the uniform asymptotic stability of the system (1.1)-(1.3) under an appropriate feedback law for {/MI, m2, m3}. From the discussion of the last section, it is reasonable to use (1.8) as the feedback law when F0 ^ 0 and (2.35) in the opposite case. Because
MINDLIN-TIMOSHENKO PLATES
41
of the length and complexity of the computations involved in deriving uniform asymptotic energy estimates for the M-T system, we shall consider only the which is the simpler of the two situations. However, in the next case chanter uniform asvmototic enerev estimates for a Kirchhoff svstem will be and when (see Chapter 4, § 6.2). derived both when In order to establish uniform asymptotic stability of the system (1.1)-(1.3) and First ol (1.8), some restrictions are needed on the geometry of all, it will be assumed thai satisfy one of the following conditions: and Theorem 2.4.
or
is a region of the type specified in
is a polygonal region of the type specified in and (2.25), (2.26) are satisfied Either of the conditions (3.la), (3.1b) will assure that if the initial data satisfy the assumptions of Theorem 2.1, the solution of (1.1)-(1.3) will satisfy
provided for some This degree of regularity is required in the derivations of the estimates below. To state our second geometric assumption, we introduce a vector field m(x, y) in IR2 defined by where {x0, }>o} is a fixed point of that
We shall assume that
and
are such
for some choice of the point {x0, y0}. Assuming (3.2), we now choose the gains /-,- in (1.8) as where
Denote by G the 3 x 3 matrix G = [gy]. We assume G is positive definite on
The following result will be required in what follows. LEMMA 3.1. Let
for some
Then
42
CHAPTER 3
In (3.5) we have written i/»x for dfy/dx, etc. Remark 3.1. Each of the integrals on the right is well defined for {(/»,
Therefore,
to prove
(3.5) we have only to transform the term We have
MINDLIN-TIMOSHENKO PLATES
43
where dX — dx dy. Also
Therefore,
Equation (3.5) now follows from (3.6) and (3.7). 3.2. An energy identity. We apply Lemma 3.1 to the solution of (1.1)-(1.3), (1.8) and we integrate that identity in t from 0 to T. We obtain
44
CHAPTER 3
All of the integrals on the left-hand side of (3.8) may be interpreted in the L 2 (Ox(0, T)) scalar product since {
where A typical term of the last integral in (3.9) is (except for a constant factor)
It follows that
Use of (3.9) and (3.11) in (3.8) gives
MINDLIN-TIMOSHENKO PLATES
45
where
Next, let us examine the integrals over F0 in the right side of (3.12). Since ^ = (p = w = 0 on r0, we have Vt/f = v(d^/dv] on F0 and similarly for V
Also,
Since
we obtain from (3.12)-(3.16)
46
CHAPTER 3
where
Next, we use Green's formula (1.7) with «A = «A>
We integrate (3.20) in t from 0 to T. After an integration by parts in the first term, we obtain
where
Let e be a positive number. Multiply (3.21) by 1-e and add the product to (3.17) to obtain
MINDLIN-TIMOSHENKO PLATES
47
The next step is to use (1.7) with $ =
where Multiply (3.24) by e and add the product to (3.23) to obtain
The identity (3.26) will be the basis for the stabilization results of the next section. Let us note that so far we have not used the geometric assumptions (3.2). 3.3.
A priori estimates. We define the function
From (3.10), (3.22), and (3.25) we see that Since (3.26) is an identity in T> 0, we may differentiate (3.26) in T. We obtain (writing t in place of T)
where the right side is evaluated at t.
48
CHAPTER 3
Set
We are going to prove that for all sufficiently small positive values of e and S,
where
Our stabilization results will then be deduced from (3.30). We begin the proof of (3.30) by estimating p'e(t). First, we estimate a\(*l>,
Since T0^0, according to Lemma 2.1 there is a constant y0 (depending on the geometry of O and the parameters \i and D) such that Therefore,
Use of (3.31) in (3.29) yields
We choose ij = |, and then choose e > 0 so that 1 - e - 2ey0K ^ e, that is,
MINDLIN-TIMOSHENKO PLATES
49
With e satisfying (3.32) we have
We proceed to estimate the last term on the right-hand side of (3.33). Since b( • ; •) is symmetric and nonnegative we have
According to assumption (3.4), there are positive constants g0 and G0 such that Therefore,
Consider, for example, the term
50
CHAPTER 3
where
Estimation of the other terms in (3.35) in the same manner yields
For K^K0>0 we have, according to Lemma 2.1 and trace theory,
In addition,
where the constants ylf j2 depend on f l , D, p, and K0, and where
From (3.34)-(3.39) we obtain the estimate
MINDLIN-TIMOSHENKO PLATES
51
Use of (3.40) in (3.33) yields
where
From the definition (3.18) of J0, (3.14), (3.16), and the first of the geometric assumptions in (3.2), we have
where (cf. (3.39))
Substituting (3.42) in the right-hand side of (3.41) gives the estimate
We choose r\ > 0 so small that
52
CHAPTER 3
(Recall that e<^.) We then obtain from (3.41)
Therefore,
where Since the gain matrix G is positive definite, there is a number g0> 0 such that
provided h2/l2^l (as we may assume). Therefore,
provided 5 > 0 is selected so that
Thus (3.30) is established. Next, we observe from the definition (3.27) of pe and Lemma 2.1 that there is a constant C such that where C depends on fl, D, /i, and K0(K^K0>Q) but not on e. Therefore,
MINDLIN-TIMOSHENKO PLATES
53
Let /3>0, multiply (3.44) by e~ftt and integrate from t to +00. We obtain after an integration by parts
Since F e S ^0 if 5C^1, we may drop the second term on the left-hand side of (3.45). If we then allow )3-»0 in what remains we obtain
where
We have proved the following theorem. THEOREM 3.1. Assume that Y0,Ti satisfy (3.1), (3.2) and that T 0 ^0. Let the gain matrix K be given by (3.3) with (3.4). Then
where e satisfies (3.32) and a) is defined by (3.46). COROLLARY 3.1. Under the assumptions of Theorem 3.1, we have
Proof. Equation (3.48) follows from (3.47) and Gronwall's Lemma. To obtain (3.49) we note that, since E ( t ) is nonincreasing, for every r>0 we have
hence,
The fraction on the right assumes its minimum at r= 1/w, and for this value of T we have Remark 3.2. From (3.32) and (3.46) we see that &>-»0 as K -»oo. However, as we shall show in Chapter 4, as K^>oo the system (2.1)-(2.3) converges to a dissipative Kirchhoff system which one can prove has a positive decay rate
54
CHAPTER 3
3.4. Optimal value of w with respect to the gains. Since thesystem (1.1)-(1.3), (1.8) is conservative when the matrix F = 0 (Neumann boundary conditions) and when F=oo (i.e., Dirichlet boundary conditions), it is natural to ask for those values of the gains which maximize the decay rate w denned in (3.46). Since o> does not depend on the individual gains gtj but rather only on the lower and upper bounds g0 and G0, respectively, we will maximize a> with respect to these quantities over the region (Kg0^G0. In addition to g0 and G0, w depends on the material parameters p, h, p, D, and K, on ft, and also on the "artificial" parameter e introduced in the course of the proof of Theorem 3.1. Therefore, we first maximize a> with respect to e (the other parameters being held fixed) and then maximize with respect to g0, G0. Let us examine the dependence of w on e. We recall that the constants e, 8 in (3.46) satisfy
where
and where Ce does not depend on g0 or G0. Since Ce is an increasing function of e for e in the interval specified in (3.50), we see that the upper bound for 77 is an increasing function of e. Also, 8 is an increasing function of 17 and the factor 5/[2(l + 5C)] in (3.46) is an increasing function of 5. It follows that, as a function of e, the largest value of CD will occur when we choose in (3.50), (3.51) the values
where C\ = C, (e) > 0 and C2 = C2( e) > 0 depend on e (and elastic and geometric constants) but not on g0 or G0. Set G 0 =rg 0 , /"=£!, so that For r fixed, we choose g0 in order to maximize 8 (and therefore ta). It is easily seen that 8 has a unique maximum at g 0 =VC 2 /> and for this g0, 8 = Ci/(2*JrC2). Finally, 8 is maximized at r = l. Therefore, the largest value of a) occurs when g 0 = G0 = -v/C^, that is, when the gain gi};=vT^ fiy, and is given by
MINDLIN-TIMOSHENKO PLATES
55
3.5. Uniform decay of solutions in other norms. 3.5.1. Strengthening the norm. Suppose that the conditions of Theorem 3.1 are met, and assume that the initial data for (1.1)-(1.3), (1.8) satisfy that is to say, {u°, u1}eD(^) in the notation of § 2.2. If u = {$,
Since [c(C~ 1 w)] 1/2 is a norm on $? equivalent to ||w||^, we have from (3.54) for some constant C0. Thus, (3.55) holds for initial data satisfying (3.53). If the initial data are smoother still, estimates in even stronger norms can be obtained. The inequality (3.55) is essentially an H2(£l) estimate for the displacement u(f) and an Hl(£l) for the velocity u'(t). Since dimension = 2, the Sobolev Imbedding Theorem gives H 2 (fl)c C(ft) with continuous injection. Therefore (3.55) implies, in particular, an exponential decay rate for displacement in the L°°(a) norm. 3.5.2. Weakening the norm. We start with data satisfying and integrate (2.3) in t from 0 to t. We obtain
Define
where v°e °V is chosen according to
56
CHAPTER 3
Then v satisfies where Av(0)+fit)'(0) = -Cu1e5if. Therefore, Av+Bv'eC([0,oo); %)• According to Corollary 3.1, we have We define a norm \\-\\x by where v0 is defined by (3.56), and we introduce the space 8?= completion ofYxffl
with respect to (3.58).
We note that (3.58) indeed defines a norm since ||{u°,u1}||^ = 0 implies u°=0, v° = 0, and then, by (3.56), that u^O. Since we have and, therefore, from (3.57) we obtain the estimate Remark 3.3. Since v(z) satisfies
u(f) satisfies
From (3.60) we see that the semigroup defined by {u°, u'}-» {u(t),u'(t)}: TxSif-^TxX is also a contraction semigroup in the topology induced by \\-\\%. Extending this semigroup from TxSif to %? by continuity, we obtain the existence and uniqueness of a weak solution of (1.1), (1.3), (1.8) for initial data {u°,ul}e%. Remark 3.4. What can be said about the space $?? Since from (3.56) we have and, therefore,
MINDLIN-TIMOSHENKO PLATES
57
Let W be the completion of Y in the norm Obviously W<= X and, from (3.61), Wx Tc % algebraically and topologically. Conversely, suppose {u°, u!}e $6. Then there is a unique v°e Y and a sequence {u°, uj,}e ^x W such that u°-»u° strongly in % and v° ^v° strongly in Y, where v ° e y is determined by A\°n = -Culn-Bu°n and satisfies, in particular, It follows from (3.62) that {uj,} converges to {L/, L2v°, L3v°} in [fr1^)]3. Therefore, Se^^x[H~l(fl)]3t at least algebraically. It would be of interest to find a more precise characterization of the space $? than that given above. Appendix. The purpose of this Appendix is to derive (2.27), whose roots determine the singular behavior of solutions of the system of equations (2.24) in a sector fl with angle a), subject to Neumann boundary conditions (2.26) on the edges of the sector and to determine the solutions of (2.27) in Re z ^ 0. Let O = {{r, #}|0< #<w, r>Q}. The behavior of {(/>,
subject to homogeneous boundary conditions
The idea is to substitute (A.I) into (A.2) and obtain a system of two secondorder ordinary differential equations for (0i(#), ®2(d)} containing z as a parameter. One then solves this system on a fundamental set of solutions. The solution will contain four unknown constants. By requiring the solution to satisfy (A.3), one obtains a linear homogeneous system of four equations for the four constants. This system has a nontrivial solution if and only if its determinant vanishes. The latter requirement turns out to be equivalent to (2.27).
58
CHAPTER 3
In order to make the computations manageable, we introduce stress resultants (as in Chapter 2, § 1.3)
Then (A.2), (A.3) may be written
Epuation J A.4) irnolies the .existence _of_a_ function. f .determined j^o.tr>_a rigid motion, such that
After an appropriate choice of the rigid motion, the function satisfies (see § 1.3 of Chapter 3)
We will determine the order rl of singularity near r = 0 in singular solutions of (A.5), (A.6). The singular behavior of {$, y} will then be in rc~l near r = 0. In polar coordinates, (A.5), (A.6) are, respectively,
We substitute into (A.7). After some computation it is found that ®(#) satisfies
MINDLIN-TIMOSHENKO PLATES
59
Let us set
The characteristic equation of (A.9) is
which has the roots Therefore, the general solutions of (A.9) is
If we impose on (A.10) the boundary conditions
we obtain the following linear homogeneous system in Cj, • • • , c4:
To obtain a nontrivial solution we set the determinant of (A.ll) equal to zero. After a little computation, it is found that the determinant vanishes if and only if z = 0 and z-l are obvious roots of (A.12) but neither leads to a singular solution. We show that there are no other roots of (A.12) in the strip 0^ Re z ^ 1 when we(0,77), or in O^Re z^\ when cue(77,277). Set z = x+iy. Equation (A.12) is equivalent to
If x = 0, the second equation in (A.13) reads sinh toy — ±y sin co. This equation has no real roots y ^ 0 since y ^ 0 implies
60
CHAPTER 3
Thus there are no roots of (A.13) on the imaginary axis, save z = 0. If x = l and y^O, the first equation in (A.13) cannot hold, since |cosho>y|>l. Thus (A.13) has no roots on the line x = l, except z = l. Case 1. 0
But (A.13) implies
Thus (A.13) cannot have roots in the strip 0
which shows that (A.15) cannot hold. Thus (A.13) has no roots in 0
Chapter 4
Limits of the Mindlin-Timoshenko System and Asymptotic Stability of the Limit Systems
1. Orientation. In this chapter we will examine the limit of the solution {i//, (p, w} = {(/fK,
Equation (1.1) is the variational form of the system
61
62
CHAPTER 4
Equations (1.2) are the classical linear dynamical equations of plane elasticity, where «/> and
We shall assume that that is, we require that initially the plate is subject only to in-plane strains and in-plane velocities. We further assume that r 0 ^0. It then follows from (2.1) and Lemma 2.1 of Chapter 3 that as K -»0, remain in a bounded set in L°°(0, T; V), remain in a bounded set in L°°(0, T; H), remains in a bounded set in L°°(0, T; V), where T>0 is arbitrary and where in (2.2) we have used the notation It follows from (2.3b) that WK remains in a bounded set in L°°(0, T; H). Therefore, from {{^K, (pK, wK}\K>0} we may extract a subnet, still denoted by {^K,
In order to obtain some sort of convergence of WK at the boundary, we assume that the gain matrix F has the form
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
63
with/ 3 3 >0 on Fj. Then from (2.1) we obtain W
'K ri*(o, D
remains in a bounded set in L 2 (F t x (0, T)).
Therefore, WK r,x(o, T) remains in a bounded set in L°°(0, T; L 2 (Fi)) so that (2.6)
WK r,x(o,D
converges weak star in L°°(0, T; L 2 (F t )) to some limit wri. A
Next, in (3.2.2) we set $ =
Passing to the limit as K -» 0, we obtain in view of (2.3c), (2.4), (2.5), and (2.6)
From (2.8) we get (as well as wr, = 0) and then from (2.2), Similarly, setting w — 0 in (3.2.2) and then passing to the limit as K -» 0, we obtain
The system (2.10), (2.11) has a unique solution with fa
64
CHAPTER 4
Assume that the gain matrix F has the form (2.5). Then as X-»0 one has weak star, weak star, weak star, where {&,
is positive definite on Yl and if (O, FO^J) satisfy (3.3.1), (3.3.2), then from Corollary 3.1, Chapter 3, we may deduce a uniform rate of decay of the energy o/(2.10), (2.11), i.e., we have
for some w 0 > 0, where a0 = (12/fc 3 )a 0 . In fact, if we denote by EK(t) the total energy of the solution of (3.2.1)-(3.2.2), we have where, in view of (2.2),
From the expression (3.3.46) for the optimal value of O)K, we see that o>fc-» tu 0 > 0 as K -» 0+. Therefore,
On the other hand, for t > 0
and therefore
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
65
Equation (2.12) follows from (2.13) and (2.14). 3. The limit of the M-T system as K -> oo. We start from (2.1) and let K -» oo. In order to obtain bounds on {tl/K(t),
that is,
In addition, we assume that
Remark 3.1. Hypothesis (3.1) means that the plate is initially free of transverse shear and is thus a requirement of consistency with the Kirchhoff model, since that model does not include transverse shear effects. Assumption (3.2) is a requirement that the initial kinetic energy associated with the Kirchhoff model is finite (see (2.2.8)). With assumption (3.1) we deduce that for any T>0, as K^oo remain in a bounded set in L°°(0, T; V), remain in a bounded set in L°°(0, T; H),
remain in a bounded set in L°°(0, T; H). It follows from (3.3b) that remains in a bounded set in L°°(0, T; H), and from (3.3a) and (3.3c) that remain in a bounded set in L°°(0, T; H). Consequently, it results from (3.3)-(3.5) that remain in a bounded set in L°°(0, T; V).
66
CHAPTER 4
Therefore, from {{
so that (3.9)
w e L°°(0, T; Hf^a)),
w' € L°°(0, T; H{-0(Q)).
It follows from (3.9) that f-» w ( f ) is strongly (respectively, weakly) continuous from [0, T] into Wj-0(n) (respectively, Hr0(ft)) (see Lions and Magenes [1, p. 275]). Since wK(0) = w°, we conclude that (It will turn out that w is strongly continuous into Hp0(il).) Next, in (3.2.2) we choose
We obtain
Using (3.7) and (3.8) we may pass to the limit in (3.11) (in the sense of distributions on (0, T)) and we obtain
LIMITS OF THE MINDLIN-TIMOSHENKO SYSTEM
67
In addition,
As we shall see in the next section, the variational problem (3.12), (3.13) has a unique solution w with
THEOREM 3.1. Let {iff0,
Let {^K,
and where w is the unique solution of (3.12)-(3.14). Remark 3.2. It follows from Theorem 4.2 that, in particular, strongly in [L2(Ox (0, T))]3. If we write out (3.12) we obtain
for every w e Hr0(n). Let us see what the corresponding boundary value problem is. We introduce a bilinear form a(w; w) on Hr 0 (H) by
and we write out Green's formula (3.1.7) in which we set
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The result is the following identity:
The first integral on the left side of (3.16) may be written
In the boundary integral in the right side of (3.16) write
where r = (-v2, ^i). Then the boundary integral may be expressed as
where
Use of (3.17) and (3.19) in (3.16) yields the following Green's formula: For all sufficiently smooth w, w,
If in (3.15) we set w = test function on fl, it follows from (3.20) that w is a solution of the Kirchhoff plate equation
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To determine the boundary conditions satisfied by w, we need to express the boundary integral in (3.15) in terms of w and dw/Bv. If we use (3.18) to replace dw/dx and dw/dy in the boundary integral and use
which is valid if w = 0 on F0, we obtain
where the "feedback operators" ^ and 2F2 are defined by
It now follows from (3.22) and Green's formula (3.20) that the boundary conditions associated with the variational equation (3.15) are
Remark 3.3. If the gain matrix F is diagonal with equal diagonal elements fu =/ 0 , then (3.23), (3.24) simplify to
4.
Study of the Kirchhoff system.
4.1. Preliminaries. We consider in this section the behavior of solutions of the Kirchhoff plate equation (3.21) under the boundary conditions (3.25),
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(3.26). The variational equation of this system is (3.15). In order to simplify the writing of the system, we make a change of the time scale t-»t\/D/ph. Then (3.21), (3.25), (3.26) is brought to the form
where ' indicates differentiation with respect to the new time variable (still denoted by t) and where we have made the notational change fi}:-» (Dp/i)1/2/i, in the gains. Introduce the forms
The variational equation for (4.1)-(4.3) may be written
Initial conditions for (4.7) are
4.2.
Existence, uniqueness, and properties of solutions. We shall set
It is obvious that c( •; • ) defines a scalar product on V equivalent to the usual one and, if F0 ^ 0, it is easy to see that a( •; • ) is equivalent to the usual scalar product on W. Let V and W be endowed with these scalar products, and define operators Ae2(W, W), Be£(W, W) and Ce2(V, V) by
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A (respectively, C) is the canonical isomorphism of W on W (respectively, of V onto V), and B is symmetric and nonnegative. Following the semigroup approach of Chapter 3 leads to the equation where
with and where ^ is the canonical isomorphism of W x V onto W x V. It may be proved as in Chapter 3 that — Vo~l si generates a C°-semigroup on contractions on WxV. We therefore obtain the following theorem. THEOREM 4.1. (i) The problem (4.7), (4.8) admits a unique solution with (ii) Assume that Then the solution w satisfies Remark 4.1. It follows from (3.16) and (3.19) that
In fact, let {w, v} e D0 and weW. Then
The last integral may be interpreted in the duality between Hl/2(T) and H~ 1/2 (F). (The quantity in brackets in the integral belongs to H~l/2(Y) since
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d/drB2(w) is a tangential derivative of a function in H1/2(F), and ^ 2 ( u ) = gi + dg2/dT with g,eH 1/2 (r).) From (4.13) we obtain
Thus, Aw + Bvc V. Remark 4.2. It is obvious that the solution w of (4.7), (4.8) satisfies the identity
if the initial data satisfy (4.11), where a(w) = a(w; w), etc. In addition,
that is,
A variant. The Case y = 0. In the special case where y = 0 in (4.1) and (4.3), the operator C is the identity map on H = L 2 (ft), and — ^^M generates a C°-semigroup of contractions on WxH, where
(cf. (4.10)). We therefore have the following theorem. THEOREM 4.2. (i) Assume that w°6 W, wleH. The problem
admits a unique solution with w e C([0, oo); W), w'e C([0, oo); //). (ii) Assume that
Then the solution of (4.17) satisfies
4.2.1. Regularity of solutions. The first question of regularity (in the spatial variables) is this: How regular is D(^), the domain of the generator of the semigroup exp (-(€'ls4t)1 To study this question, consider the equation
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The precise regularity possessed by solutions of (4.19) will be difficult to determine in general, but it is reasonable to expect that {w, V}E D0 (defined in (4.13)) if r 0 nr 1 = 0 and F is sufficiently smooth. (This will certainly not be true if r o n f\ ^ 0.) In particular, one may expect that w e H 3 (fl). We are, however, unable to establish this property except in the special case y = 0. (Note that V is not a space of distributions on H when Yl ^ 0.) In this particular situation, in place of (4.19) we have
It may be seen with the aid of Green's formula (3.20) that (4.20) is the variational form of this system
When r o nF 1 = 0 and F is smooth, (4.21) is a regular elliptic boundary value problem and the regularity of w can be deduced from classical elliptic theory as presented, for example, in Lions and Magenes [1, Chap. 2]. Following that approach, let £(X) G 2(Cl) be positive on fi U F0 and vanish on Yl at the same rate as d ( X , F^, the distance of X to F! :
For m =0,1, • • • , define the space
One has Hro(O)c: S|"o(n)cr L 2 (H) with continuous injections. Furthermore, it may be shown that ®(O) is dense in Sr0(^) and therefore Hr o m (O), the dual space of Hr 0 (O), is a space on distributions on (1 with L 2 (H)ci SF0m(O)c //~ m (fl) algebraically and topologically. For m = 0,1, 2, 3, let the space Dr o (H) be defined as follows:
Consider the map
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The same arguments as put forth in Lions and Magenes [1] shows that (4.22) defines an algebraic and topological isomorphism o/D™0((l)\^V" onto Er 0 ~ 4 (fl) x Hm-s/2(rl)xHm-1/2(rl), where
(If r 0 /0, then ^" = {0}.) In particular, it follows that any solution of (4.21) belongs to £>ro(ft) <=• Hra(Q), the embedding being continuous. As a consequence, we have the following result. THEOREM 4.3. Assume that y = Q, that T is smooth and r o nr, = 0. Let w be the solution of (4.17) with initial data satisfying (4.18). Then
Remark 4.3. In the situation of Theorem 4.3, assume that in the boundary operators ^, and 3Sf2. Then the boundary conditions in (4.21) satisfy
so that the solution of (4.21) has the regularity w6H 4 (fl). Therefore, when (4.24) holds, the solution of (4.17) with initial data given by (4.18) satisfies
Let us return to the case y ^ 0, still assuming that F0 ("I f\ = 0. Consider the equation 3${w, v} = {g,f}, that is, If {&/}e W' x W, then (4.25) has a unique solution {w,v}eWxV given by
for some C > 0. Therefore,
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On the other hand, if {g,/}e W'x H, we obtain (4.26) for v and (4.20) for w. It follows from the discussion following (4.20) that w e H 3 (ft),
Therefore, We may now interpolate between (4.29) and (4.28): We have (see Lions and Magenes [1])
We use (4.31) with d = \ and obtain
It follows that We have therefore proved the following result. THEOREM 4.4. Assume that T is smooth and F0 D f\ = 0. Let w be the solution of (4.7) with initial data {w°, w1} satisfying (4.11). Then w satisfies
In particular, it follows from Theorem 4.4 that the resolvent of —^~^si is a compact operator on W x V. Remark 4.4. In the context of Theorem 4.4, if (4.24) is also satisfied we have w e C([0,oo); H3((l)) since, in this situation, d~le£(W'xH, (// 4 (H) 0 W) x W), and [H4(H) fl W, W]l/2 = H3((l) H W. In order to obtain more regularity of the solution than is given by Theorem 4.4, we need to require more regularity of the initial data than is supposed in (4.11). The requisite assumptions are obtained as follows. We introduce and we assume
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From (4.12) we have Cw"(0) = -(Aw°+Bwl)e H and, therefore, We further assume Then we may conclude that Indeed, from (3.16), (3.19), and (3.22) we have
If (4.34) holds, then
Consequently, (4.35) holds as claimed. We have shown that if (4.33) and (4.34) hold, then
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that is, {w1, w"(0)} e D(s&}. It follows from Theorem 4.4 (since w' is a solution of (4.7) with initial data {w1, w"(0)}) that w satisfies
Then, as a consequence of Aw + Bw' = - Cw" (t ^ 0) we obtain that w satisfies
Elliptic theory then gives
Remark 4.5. The conditions in (4.33) will be met if the initial data satisfy
In fact, these conditions imply that
(that is, Aw°+BwleH), as may be seen with the aid of Green's formula (3.20). In particular, {w°, w1} e D(M] in view of (4.13). From Theorem 4.1 and (4.40), we have
Equations (4.40) and (4.41) imply that w"(0) is a weak solution of the problem
Since F o nr i = 0, (4.42) gives w"(0) e H2(O). But since w'eC([0,oo); W) we have dw'(t)/dv = Q on T0 for r ^ O , from which it follows that dw"(Q)/dv = Q on T 0 . Therefore, w"(0) e W so that Aw° + Bw1 = -Cw"(0) e H.
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Finally, we give some conditions on {w°, tv1} which will assure that (4.1)(4.3) has classical solutions. Let the initial data satisfy (4.33), (4.34). If, in addition, the data is such that (w"(0), w'"(0)}e D(sd), then w' will have the regularity C([0,o>); H 7/2 (ft)) and w"e C([0,oo); H5/2(fl)). Consequently,
The argument leading to Theorem 4.3 then gives
Sufficient conditions on {w°, w1} that guarantee that all requirements are met are as follows: {w°, w1} satisfy the compatibility conditions (4.39) and also the compatibility conditions
In fact, (4.43), (4.39), and the first condition of (4.44) imply that (4.33) and (4.34) are satisfied. Moreover, following the argument in Remark 4.5, we obtain from (4.43) and (4.44) that w"(0)e H 7/2 (O) (since \2w°e H\fl)), w'"(0)e Hs/2(fl), and dw'"(Q)/di> = 0 on F,. Then we may use (4.45) to show that Aw"(0) + Bw'"(0) e V. Therefore, {w"(0), w'"(0)}eD(^). 4.2.2. Strong asymptotic stability of solutions. We consider the system (4.1)-(4.3) and assume that T is smooth and F o nr i = 0. Then -^si has compact resolvent, by virtue of Theorem 4.4, and therefore strong asymptotic stability may be proved as in § 2.4 of Chapter 3. Indeed, everything comes down to showing that
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But (4.48), (4.49) imply1
and then, since solutions of (4.47) are (real) analytic in H, that w = 0 in O. We may therefore conclude that for every solution of (4.7) in the space Wx V. If F0 = 0, then A = 0 is an eigenvalue of <#~1.s# and (4.50) is not true for all finite energy solutions of (4.7), regardless of how the gain matrix is chosen. We may eliminate the eigenvalue at A = 0 by working in the space WxV, where
Then a(w; w) (respectively, c(v; v)) is strictly coercive on W (respectively, on V) and (4.50) is valid for solutions of (4.7) with initial data {w°, w1} e W x V. We may achieve strong asymptotic stability in the space W x V when F0 = 0 by a suitable modification of the feedback law (4.3), as in § 2.4.2 of Chapter 3. With this in mind, we replace (4.3) by
<*
*<
We suppose that ^(w) and ^ 2 (w) have the same structure as ^i(vv) and ^ 2 (w), respectively, (cf. (3.23), (3.24)). Then
*r
for some symmetric 3 x 3 matrix F. We introduce a bilinear form d(w; w) on W defined by
1
We may write
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The key point is the following: If F is positive definite on a set in F of positive measure, then a(w; w) is strictly coercive on W. The proof is similar to that of Lemma 2.3 of Chapter 3. The variational equation corresponding to (4.1), (4.2), (4.51) is
Equation (4.54) is well set in the space Wx V. Furthermore, solutions of (4.54) are strongly stable in W x V, that is, provided (i) F is positive definite on a nonempty, open subset of F; and (ii) F is positive definite on a subset of F of positive measure. The proof is essentially identical to that given for the case F0 ^ 0. 5. Uniform asymptotic stability of solutions: The case F0 it 0. 5.1. Geometric assumptions. As in Chapter 3, § 3, we can establish uniform asymptotic stability of solutions of (4.1)-(4.3) only under additional assumptions on the geometry of O and the gain matrix F. To simplify matters a bit, we will assume that F is smooth. Furthermore, we will require Equation (5.1) is imposed in order to assure that classical solutions of (4.1)(4.3) will exist if the initial data are appropriate (see § 4.2.1). Also, analogous to (3.3.2) and (3.3.3) we assume
for some choice of {x0, y0} e IR2, where m{x, y} = {x, y}-{x0,y0}, and where g{J e C^Fj), and where is positive definite on Fj. We shall need a formula analogous to (3.3.5). LEMMA 5.1. Let we H x (fi)/or some s>l, s^4. Then for all //.eR
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Remark 5.1. The integral on the left-hand side of (5.5) may be interpreted in the duality between Ho~s(£l) and Hs~4(£l) since m - V v v s H'-l(£l)cH4-s(£l) = H%-°(£l) provided s>7/2, s^4. Proof. It suffices to establish (5.5) for w e //4(O) since each integral in (5.5) has a sense if w e Hs(£l) and depends continuously on w in that space. Therefore, suppose w e H4(O) and apply Green's formula (3.20) with vv = m • V w. We obtain
Let us calculate a(w; m - V w ) . We have
When the differentiations of (m- Vw) beneath the integral are carried out, one finds that a(w\ m - V w ) may be written
Inserting (5.7) into (5.6) yields (5.5). 5.2. An energy identity. We assume that the initial data {u>°, w1} are such that the solution of (4.1)-(4.3) with data {vv°, w1} is classical:
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for some s > |. Conditions that assure this degree of regularity are given in §4.2.1. We apply (5.5) with w the solution of (4.1)-(4.3) and integrate that identity in t from 0 to T. We obtain
The first term in (5.8) may be written
where Let us next examine the integrals over F0 in (5.8). Since w = dw/dv = Q on F0 we have Bt w = B2w = 0 there and
since wxxwyy - w2xy = 0 on F0 (as a consequence of the fact that Vw x | ro and VWj,| ro are linearly dependent). Therefore, from (5.8)-(5.11) we have
Calculation of
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Substituting (5.13) into (5.12) yields
Next, we substitute w = w into Green's formula (3.20) and integrate over [0, T]. With the aid of (3.22) we obtain
The first term on the left may be written
Therefore, (5.15) may be rewritten as
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where Multiplying (5.16) by \ and subtracting the result from (5.14) yields
where
is the total energy of the system. Equation (5.17) is the identity from which exponential decay of solutions will be deduced. We note that (5.17) does not require that the geometric assumptions (5.2a), (5.2b) be satisfied. 5.3.
A priori estimate. We define the function
Then We differentiate (5.17) in T and write t in place of T. The result is
where the right-hand side is evaluated at t. We proceed to estimate the term b(w'; m-Vw).
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We have
Let g0 > 0 and G0 > 0 be the largest and smallest numbers, respectively, such that
Since a(w; w) is strictly coercive on Hr0(O), we have from (5.21), (5.22), and trace theory
where A 0 is a constant depending on O and /x. (The last estimate uses 0 < ^ < 1; in fact, O < / A <\ in the physical situation.) Use of (5.24) in (5.21) gives
Use of (5.25) in (5.20) yields
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We choose 17 >0 so small that 2i7G0A0^|, that is,
Then from (5.26) and assumption (5.2b) we obtain the estimate
where
in view of (5.11) and the geometric assumptions (5.2a), (5.2b). Let e > 0 and introduce the function From (5.28) we have
But
provided that y = h2/l2^l, as we may assume. Use of (5.30) in (5.29) yields
provided 1 - e/2rj > 0. We choose e so small that
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that is,
Then
where
The estimate (5.33) is analogous to (3.43) of Chapter 3. From the definition (5.18) of p ( t ) we see that
for some constant C depending on ^ and fl but not on y if 0< y ^ 1. We may now proceed exactly as in Chapter 3 and conclude that
with
As a consequence of (5.35) we obtain the estimates
Let us note that
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THEOREM 5.1. Assume r 0 ^0, that F0, Yl satisfy (5.1), (5.2), and that the gain matrix F satisfies (5.3) and (5.4). Then the estimates (5.36)-(5.38) hold for every solution of (4.1)-(4.3) for which {w(0), w'(0)}e //ro(H) x Hj.Q(ft). Remark 5.2. As in Chapter 3, § 3.4, one may maximize a with respect to e and the bounds g0, ^o on the gains. Remark 5.3. One may obtain estimates for the solution in other norms, analogous to what was done in Chapter 3, § 3.5. Remark 5.4. All of the estimates leading to (5.35) remain valid if the matrix G satisfies
with go > 0 and independent of y. The significance of this remark will be seen in the next section, where we allow y-»0+. 6. Limit of the Kirchhoff system as y -» 0+. Let wy denote the solution of (4.7), (4.8). From (4.7) we obtain
provided b(w'y)eLl(Tl x(0, T)) (which will be the case if, for example, {w°, w1} satisfy (4.11)). We assume that r 0 ^0. It follows from (6.1) that as y->0+ remains in a bounded set reamins in a bounded set o/ remains in a bounded set o/ where we recall the notation
Therefore, choosing a subnet of {wy \ y > 0} (still denoted by {vvy}) we obtain
From (4.7) we have
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Using (6.2c), (6.3), and (6.4) we may pass to the limit as y-»0+ in (6.5) (in the sense of distributions on (0, T)) and we obtain
In addition, If {w°, vv^e Wx H, the problem (6.6), (6.7) admits a unique solution with Equation (6.6) indicates that w is a weak solution of the "classical" Kirchhoff plate equation and satisfies the boundary conditions
Remark 6.1. The above convergence result tacitly assumes that the form b(w\ w) is independent of y. In the context of Remark 5.4, let us assume that the matrix G satisfies (5.39), so that b = by. We then obtain from (6.1) remain in a bounded set of and, therefore, remain in a bounded set of Then, passing to the limit in (6.5) as y-»0+ through an appropriate subsequence yields
where
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/33 = (w v)g33. The variational equation (6.11) signifies that w is a solution of the plate equation (6.8) with the Dirichlet boundary conditions (6.9) on r0 x (0, oo) and
6.1. Uniform asymptotic stability of the system (6.8)-<6-10): The case T0 ^ 0. As noted above, the decay rate w obtained for the energy of the system (4.7), (4.8) (under the assumption that r 0 ^0) is independent of y (for y ^ l ) . Consequently, we may easily prove the same decay rate for the energy of the limit system (6.8)-(6.10) (or the system (6.8), (6.9), (6.12)). In fact, as 7^0+ we have for t ig 0 weakly in W, weakly in H, and, therefore,
The estimate (6.15) is valid provided r 0 ^0, T 0 , Fj satisfy the geometric assumptions (5.1), (5.2), and (i) the matrix G satisfies (5.3), (5.4), or (ii) G satisfies (which give the boundary conditions (6.12)). In the latter situation, uniform asymptotic stability was previously established by a direct argument in Lagnese [3]. 6.2. Uniform asymptotic stability of solutions of (6.8): The case T0 = 0. We consider the classical Kirchhoff equation (6.8) with boundary conditions
Based on the discussion of § 4.2.2 regarding strong stability of solutions when r0 = 0, we define v} and v2 by the feedback laws
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We assume that F is star-shaped with respect to some point X0 e [R2, that is, Furthermore, to simplify the discussion we choose where m(X) = X-XQ, and we choose the matrix F associated with ^, &2 (see (4.48)) to be diagonal with diagonal entries a(m-i>), a(m- *>),and)8(m-i>), respectively, where a, ft, and A are positive constants. The boundary conditions (6.16) then take the form
From (4.48) we have
6.2.1. An energy identity. To obtain uniform asymptotic energy estimates for (6.16), (6.18), we proceed along the lines of § 5. We start with initial data {w°, wl}e H4(fl) x H2(O) that satisfy the boundary relations (6.18) with w, w' replaced by w° and w 1 , respectively. Since F0 = 0, the solution w will be classical: w e C([0, oo); H4(O)), w' e C([0, oo); H2(H)), w"e C([0, oo); L2(H)). We apply the identity (5.5) to this solution and integrate (5.5) in t from 0 to T to obtain
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where ar(w) is defined by
Integrating by parts in t in the first term on the left and rewriting the last term on the right using (6.19), we obtain
where By the same computation as in (5.13), we have
which, when substituted into (6.22) yields
Next, substitute w = w into Green's formula (3.20) and integrate over [0, T] to obtain
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An integration by parts in the first term on the left allows (6.25) to be rewritten
where
Multiply (6.26) by ^ and add the product to (6.24) to obtain
Uniform asymptotic energy estimates will be deduced from (6.28). 6.2.2.
A priori estimates. Define the "energy" E ( t ) by
and set
where e > 0 and
From (6.23), (6.27) we have It is clear that
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for a suitable constant C which depends on ft, p, a, /3, and A. We wish to establish the estimate
for some positive constant c and for all e < e0 for some e0> 0- It will be proved below that (6.34) is indeed true provided also that a and ft are sufficiently small. More precisely, it will be proved that there is a number a0 > 0, and positive numbers j80 and e0 depending on a, such that (6.34) holds if a g a 0 , p^/30, and eg e0. To establish (6.34) we first calculate p ' ( t ) using (6.28) and (6.32). The result is
Let us calculate the term (writing {x, y} = {x^, xj and using standard summation convention)
Using (6.36) in (6.35) leads to
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We estimate the last three terms on the right-hand side of (6.37) as follows. For arbitrary S > 0, 17 > 0, and cr > 0 we have
where the C/'s are independent of the parameters 5,17, a: Using (6.38)-(6.40) in (6.37) yields the estimate
We choose
Then
Next, choose
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Then if 0 < a S a 0 a n d 0
Therefore,
provided e > 0 satisfies
where
The estimate (6.45) is analogous to the estimate (3.43) of Chapter 3, and by the same procedure employed there we obtain the following uniform asymptotic energy estimates. THEOREM 6.1. Assume that F is smooth and star-shaped with respect to some point in R2. Let w be a solution of (6.8), (6.18) with (w(0), w'(0)}e H2(fl) x L 2 (ft), where a >0, /3 >0, and A >0. There is a number a0>0 and, for each a, a number /3 0 («) > 0 such that ifa^a0 and j8 ^ /30(a) the following estimates hold:
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for some u> > 0. Remark 6.2. ta is given by
with C defined by (6.33) and where e satisfies (6.46). From (6.38) and (6.39) it is easy to see that the constants C, and C2 in (6.46) and (6.42), respectively, may be chosen as
Remark 6.3. The requirement of Theorem 6.1 that a and /3 be small is probably an artifact of the proof and not essential to the validity of the estimates.
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Chapter 5
Uniform Stabilization in Some Nonlinear Elastic Plate Problems
1.
Uniform stabilization of the Kirchhoff system by nonlinear feedback.
1.1. Setting the problem. We study the question of uniformly stabilizing the solution w of a Kirchhoff system by means of nonlinear feedback laws for the controls u, and v2 acting in the two boundary conditions on F,. To simplify matters, we will consider only the model equation (4.6.8), and we assume that the bending moment vt on r\ vanishes (cf. (4.6.12)). Thus the problem to be considered is
The energy of the system (1.1)-(1.3) is where a(w) = a(w; w), and where a(w; w) is given by (4.4.4). By Green's formula (4.3.20)
We set 99
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where / is a single-valued, real continuous function defined on U. Then
Let B be the nonlinear operator from W = Hr0(Q) into W defined by
Note that since dimension = 2, Hr0(il)<^C(Ci) with continuous injection (under mild conditions on ft) and therefore f(w(X)) is continuous on Tlt so that Bw is a well-defined element of W for each w e W. Moreover, the map w -> Bw is continuous from V^ into W. In addition, the operator B is monotone if and only if / is nondecreasing, since
If / is continuous and nondecreasing on R and /(O) = 0, the problem (1.1)-(1.4) may be solved by nonlinear semigroup methods. Indeed, we may proceed as in § 4.4 of Chapter 4 and write (1.1)-(1.4) in the form where
with One verifies that si is maximal monotone on WxH. In fact if U t = {M, , uj and Uz — ("2. v2} we have
so that s4 is monotone. To see that Range (I + sg)=WxH, let {f, g}e WxH and consider the equation that is,
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Inserting «=/+i> into the second equation in (1.10) gives Since A is strictly coercive and (Bv, v)^0 on W, it is easily seen that (A + B + IY1 (defined on Range (A + B + I)) maps bounded sets in W to bounded sets in W. Since, moreover, A + B is monotone and continuous from Winto W, it follows that Range (A + B+I)= W (see, e.g., Browder[l, Chap. 1]). If v solves (1.11) and u =f+v, then from (1.11) we have Au + Bv = g~ve H. Thus{u, v}zD(d). Since s£ is maximal monotone, it follows that — si generates a continuous semigroup of (nonlinear) contractions on D(s4). Therefore, if [w°, w1} e D(stf), there is a unique function U = {H>, w'} which satisfies t^{w(t), w'(t)} is Lipschitz continuous from [0,oo) into WxH; w'(t)eW, Aw(t) + Bw'(t)eH, t>0; t^{w(t), w'(t)} is strongly right differentiable and weakly differentiable from (0,oo) into WxH; t-> Aw(t) + Bw'(t) is weakly continuous and strongly right continuous from (0, oo) into H;
w"(t) + (Bw'(t) + Aw(t)) = 0,
t>0.
For proofs of the above properties see, e.g., Brezis [1, Chap. 3]. Remark 1.1. Let us assume that/(O) = 0. Then D(jit) is dense in WxH. In fact, suppose {w, v}£ D(st). Then we W, v e W, and for some F e H. Since/(0) = 0, (1.12) signifies that w is a weak solution of the boundary value problem
Since/(0) = 0, it follows from (1.13), (1.14) that, in particular, D(s4) => W 0 x Ho(n), where we W0 means that w e //4(O) fl Wf/fl) and satisfies homogeneous boundary conditions
But W0x H20(£l) is obviously dens_e in Hr 0 (fl) x L2(fl). Remark 1.2. Suppose that r0 H Tj = 0, /(O) = 0, and / is differentiable with /'eLJUR). Then {w, v}<=D(s&) implies we// 4 (O). Indeed, f(v)\rieH^TJ and so, by elliptic regularity theory, solutions of (1.13), (1.14) satisfy w e H 4 (fl). 1.2. Uniform energy estimates for solutions of (l.I)-(l-5). In this section we shall assume that T0 and F, satisfy (4.5.1) and (4.5.2). It is also assumed that F 0 ^0 (however, see Remark 1.6 below) and that the function/ in (1.5) has
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the form (1.15)
/=(«•")&
where g is a differentiable, nondecreasing function with g'(£) bounded on bounded sets of f and with g(0) = 0. In addition, g is required to satisfy the following growth conditions:
for some c0 > 0, p =£ 1; and for some C0> 0 and q § 1. Under these conditions we shall prove the following theorem. THEOREM 1.1. Let {w°, wl}e WxH and w be the solution of (l.l)-(l.S). There is a constant 0 such that
COROLLARY 1.1. (a) Ifp = l, (b) Ifp > 1, there is a constant A > 0 depending on E(0) such that
Remark 1.3. The constant
SOME NONLINEAR PROBLEMS
where
The left-hand side of (1.21) may be written
where Use of (1.22) in (1.21) allows (1.21) to be rewritten
The integrals over F 0 x(0, T) combine to give (see (4.5.11))
Therefore, (1.24) may be written
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1.2.1. A priori estimates. Introduce so that y, = p(T)-p(0) from (1.23). Since T 0 ^0, evidently for an appropriate constant C. From (1.25) and assumption (4.5.2) we obtain
For e > 0 we define We are going to derive the estimate Henceforth we shall assume p> 1. (For p = 1 the calculations are similar, but simpler.) We then have
Use of (1.27) and (1.28) in (1.30) gives
Let us estimate the last term in (1.31). We have
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Substituting (1.32) for the corresponding term in (1.31) yields
(1.33)
Introduce the sets (for t fixed) We estimate the last term in (1.33) as follows:
Since g' is bounded on bounded sets and g(0) = 0, there is a constant M such that Therefore,
As for the second term on the right-hand side of (1.34), we apply the inequality
with
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We obtain
Assumption (1.17) implies
Use of (1.37) in (1.36), combined with (1.35), results in the estimate
By the Sobolev Imbedding Theorem we have H 1/2 (F)c L r (D with continuous injection for every r> 1. Therefore,
We apply (1.39) to the first two terms on the right-hand side of (1.38) and obtain
It follows from (1.38) and (1.40) that
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Replacing the last term in (1.33) by (1.41) gives the estimate
for some constants C, C, and C2. The last term in (1.42) is bounded above as follows. We have from (1.16)
Also, by Holder's inequality
Use of Holder's inequality once again gives
for some constant C4. Therefore, from (1.43), (1.44) we have
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Inserting the last inequality back into (1.42) gives an estimate of the form
for some constants C, through C4 and for e > 0, S > 0, and, say, 8^1. Choose 8 > 0 so small that
Then choose e > 0 so small that
We then obtain from (1.45)
as claimed. Let )3>0, multiply (1.46) by e'1", and integrate from t to oo. After an integration by parts we obtain
In view of (1.27) and (1.29) we have
Therefore, F E ( f ) g O if
Letting /3^0 in (1.47) and using (1.48) yields
where
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Proof of Corollary 1.1. We consider only the case p> 1. From (1-46), (1.48) we have
where
Upon integrating (1.50) we obtain (with a = ( p - l ) / 2 )
provided F E .(0)>0. Assume that
Then
It follows from (1.51) and (1.52) that
where
Remark 1,5. The calculations leading to the estimate (1.18) do not utilize the assumption that g ( - ) is nondecreasing. Indeed, everything formally goes through assuming that g ( - ) satisfies (1.16), (1.17), is locally Lipschitz continuous, and g(0)=0. On the other hand, the growth restriction (1.17), which was essential to the proof, should be unnecessary. An interesting question is whether utilizing monotonicity of g in a suitable manner would allow (1.17) to be dropped. Remark 1.6. Stability results of a different nature, using nonlinear feedback at the boundary, were obtained by Wang and Chen [1] in the case of the one-dimensional wave equation. Their results were greatly generalized by Lasiecka [1] to include a variety of second-order systems of Petrowsky type, including wave equations in R" and the plate equation (1.1), and a number of different possible boundary conditions. In both papers, monotone and possibly multivalued feedbacks/fw') (important for applications) were considered. The point of both papers is to characterize the w-limit set of the initial data and to establish strong convergence of solutions to the cu-limit set.
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Remark 1.7. Although we have assumed that r05* 0, the opposite case may also be handled by changing the boundary conditions (1.3), (1.5) to
and combining the techniques of § 6.2 of Chapter 4 with those of the present section. In this way one obtains estimates of the form (1.18)-(1.20) for the "augmented energy" E ( t ) defined in (4.6.29). 2.
Uniform asymptotic energy estimates for a von Karman system.
2.1. Setting the problem. We consider in this section the von Karman system (2.1.26)-(2.1.28). Upon making the change t^t^D/ph in the time scale and the notational change F -» DF, this system is brought to the form
where y = 2D/(Eh) and where the bracket [i/>, tp] denotes the bilinear expression The boundary conditions associated with (2.1) are taken to be
(Thus we assume that F0 = 0 and that Fj = F, referring to the notation used throughout these notes.) The initial conditions are No initial conditions are prescribed for F. As usual, v1 and v2 are the controls through which the energy of the system (2.1)-(2.5) is to be stabilized. For this system the energy is defined as
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The first two terms on the right-hand side are, respectively, the kinetic and strain energies of the plate in bending in the Kirchhoff theory. The remaining term is due to the fact that in-plane stress resultants contribute to the total energy in large deflection theory. Let us calculate £'(0- We have
Using Green's formula (4.3.20) and (2.1) we obtain
the last equality being due to the following lemma. LEMMA 2.1. The trilinear form ty,
Symmetry of the form ([i/f,
where a, /3, and A are positive constants, and where we assume that F is star-shaped with respect to some point X 0 eR 2 : From (2.7) we then obtain
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Therefore, the system to be studied consists of (2.1) together with initial conditions (2.5) and boundary conditions
2.2. Variational formulation and existence of solutions. We introduce a bilinear form d(w; w) on H2(£l) by
where a(w; w) is given by (4.4.4). We recall that d(w; w) is strictly coercive on H2(£l). The variational form of (2.1), (2.5), (2.10) is as follows: Given w°e H 2 (n), w1 e L2(O), find w, F such that for every T> 0,
Remark 2.1. From (2.11) it follows that
Since the dimension is two, we have (Peetre [1])
Therefore, the trilinear terms in (2.13) and (2.14) have a sense (recall that (O, F], w) = (|>, w], F)). It also follows from (2.14) and (2.15) that w"eL°°(0, T;(H 2 (a))') (since Ll(fl)c (H2(fl))'), hence t^w'(t) is weakly continuous into L 2 (ft) (Lions and Magenes [1, Lemma 3.8.1]). Therefore, (2.12) has a sense. Moreover, from (2.11), (2.13), (2.15), and (2.16) we have
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and therefore
if e S 1, e ?4 (Lions and Magenes [1, Thm. 2.8.3]). THEOREM 2.1. // {w°, w1} e H2(fl) x L2(O), there exists a solution of (2.11)(2.14). Remark 2.2. Solutions of (2.11)-(2.14) are not, in general, unique (see Knightly and Sather [1]). The proof will be omitted since it is more or less identical to that given in Lions [1, Chap. 1, § 4], where the von Karman system (2.5) is considered with Dirichlet boundary conditions on both F and w. Remark 2.3. Regularity of solutions is (apparently) an open question. More precisely, what are conditions on the initial data {w°, w1} that guarantee that every solution of (2.11)-(2.14) is classical, that is,
If (2.17) should hold, then F is even more regular:
Inclusions (2.18a), (2.18b) are arrived at in the following manner. One has (Peetre [1])
and therefore
which implies (2.18a). Moreover,
which implies (2.18b). (For regularity of solutions of the stationary von Karman system, see Lions [1], or Ciarlet and Rabier [1].) 2.3. An energy identity. Let w, F be a solution of (2.1), (2.10) which has the regularity (2.17). We apply (4.5.5) to w and integrate the resulting identity in t from 0 to T. The result may be written
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By what is now a familiar calculation, we have
where Next, let us calculate the expression ([w, F], m • Vw) that appears in (2.19). According to Lemma 2.1, we may write But
Therefore, from (2.22), (2.23), and (2.1) we obtain
We may apply the identity (4.5.5) with p = 1 to the first term on the right-hand side of (2.24). Since F = dF/dv = 0 on T we have m - V F = 0 and d(m • VF)/dv = (m • »/)AF on T and so
On the other hand,
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Substitution of (2.25) and (2.26) into (2.24) yields
Use of (2.20) and (2.27) in (2.19) results in the identity
Let us next apply Green's formula (4.3.20) with w = w. After an integration in / from 0 to T we obtain
The first term on the left-hand side of (2.29) may be rewritten as
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Substituting (2.30) for the corresponding term in (2.29) gives
where
Multiply (2.31) by \ and add the product to (2.28) to obtain
2.4. A priori estimates. Having established the identity (2.33), the derivation of suitable a priori estimates proceeds along the lines of § 6.2.2 of Chapter 4. One introduces p(t) by
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so that
We also introduce
where E ( t ) is defined by (2.6), and we set
From (2.9) we have
and from (2.33) we have
The last three terms are bounded from above using (4.6.38)-(4.6.40). The following estimate is thereby obtained:
where ij,
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0 < /3 s a/(8C 2 ). Then choose 77, a; and S according to (4.6.42). We then obtain
It follows from (2.37) and (2.38) that
provided e >0 is chosen according to (4.6.46), where
As a consequence of (2.39) we obtain the following result. THEOREM 2.2. Assume that T is smooth and star-shaped with respect to some point in IR2. Let w, F be a solution of (2.5), (2.10) which is regular in the sense of (2.17). There is a number a 0 >0 and, for each a, a number /8 0 (a)>0 such that, if 0 < a § a 0 , 0 < /3 ^ /30(a), and A > 0, the following estimates hold:
for some 0.
Remark 2.4. The smallness of a and /? is probably inessential to the conclusions of Theorem 2.2.
Chapter 6
Asymptotic Energy Estimates for Kirchhoff Plates Subject to Weak Viscoelastic Damping
1. Formulation of the boundary value problem. We consider the system (1.36)-(1.38) of Chapter 2 which, we recall, was obtained under the assumption that the viscoelastic Poisson's ratio /t is constant. In order to simplify the writing, we make the change t -» f\/D(0)/p/i in the time scale. Equation (2.1.36) is then brought to the form
where the kernel D(t) satisfies D(0) = 1, and the boundary conditions (2.1.37), (2.1.38) become
In (1.3), we have introduced boundary operators Sli, 8&2 defined by
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where Bl and B2 are denned by
"Initial" conditions for the system are The functions Vi, v2 are the controls. We shall assume that the function D satisfies Hypotheses (1.5) assure that the viscoelastic energy (denned below) is nonincreasing, that Dx = D(<x>) and D'x = D'(oc) both exist, that D^Q, and D'x, = 0. We shall suppose that Assumption (1.6) means that the material behaves like an elastic solid at t = +00. We now seek feedback laws for the controls vl, v2 that will induce further dissipation in the system. First, the "energy" of the system must be properly denned. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression, we first rewrite (1.1) and (1.3) in the form
where in (1.7) and (1.8) we have introduced the notation w,(s) = w(t-s), We multiply (1.7) by w' and integrate the product over ft. With the aid of Green's formula (4.3.20) we obtain (proceeding formally; everything will be justified in the next section)
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
We have
The first term on the right-hand side of (1.10) equals
The second term on the right-hand side of (1.10) equals
Therefore,
Consequently, we have
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Substituting (1.11) into (1.9) and using the boundary conditions (1.8) yields
We define the total energy of the system (1.1)-(1.5) as
Then (1.12) may be written
The last term on the right-hand side of (1.14) represents viscoelastic damping. We may introduce additional damping through an appropriate choice of u, and v2 in the first integral on the right-hand side of (1.14). It is reasonable to choose Vi and v2 as we did in Chapter 4, § 3: where ^i,^ 2 are defined in (4.3.23), (4.3.24), and where the gain matrix F = \-ftj] is positive semidefinite on F,. We then have
Therefore, the initial-boundary value problem to be considered is (1.1)-(1.4) with the feedbacks v\, v2 defined by (1.15). 2. Existence, uniqueness, and properties of solutions. 2.1. Function spaces and variational formulation. We will adhere to the definitions of Chapter 4, § 4 of the function spaces W( = Hr0(fl)) and
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123
V( = Hf- 0 (ft)) and the bilinear forms a(w; w), b(w; tv), and c(w; w) (see (4.4.4)_(4.4.6)). We also define the Hilbert space of W-valued functions
Let aK denote the product space normed by The space %t is a "finite energy space" of the type first introduced for viscoelastic problems by Dafermos [2]. We define an operator L<E%(W, W) by
where the integral is a Bochner integral in W. For a sufficiently regular function t-*w(t): (-oo;oo)-* W of, say, compact support, we can define a function t-» w,: (0, oo)-» W by setting With the above definitions, we may now give a variational formulation of the initial-boundary value problem (1.1)-(1.4), (1.15) as follows: Find a W-valued function w defined on (—00, oo) such that is continuous;
given in Equivalently, we may replace (2.7) by
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which amounts to writing (1.1), (1.3) in the form (1.7), (1.8). Equation (2.9) is in turn equivalent to
2.2. Well-posedness of (2.5H2-8). The problem (2.5)-(2.8) may be solved by a semigroup approach similar to that employed for the M-T system and for the elastic Kirchhoff system. First we note that off', the dual space of S€, is Let AW denote the canonical isomorphism of W onto W. It is denned by (Awii)(s) = A(u(s)), w e W. We introduce and define the linear operators
with
<£ is the canonical isomorphism of 3€ onto W and <
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125
(i) jrf is densely defined in 3C since D(s$) contains all {w, v, u( •)} e % which satisfy
(ii) —<€ ^si is dissipative with respect to ( • , • ) * • since for {w, v, u( • ) } e D(jt),
(iii) Range (/+ < ^ > ~ 1 J^) = 5if. To prove (iii), let {/, g, h( •)}e S€ and consider the equation Equation (2.13) is equivalent to
From (2.14c) we obtain
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where w is yet to be determined. We show that ueW provided we W. For fixed s0 > 0, we calculate
therefore,
It follows that
hence u e W and then (du/ds)eW from (2.14c). Next, we replace u and v in (2.14b) using (2.14a) and (2.15) and obtain the following equation for w:
where
Since Dx > 0, it is easily verified that
Since the right-hand side of (2.16) belongs to W, it follows from the LaxMilgram Theorem that (2.16) is uniquely solvable for weW. Finally, ve W is determined from (2.14a) and w e W is defined by (2.15). 2.3. Regularity of solutions. We consider only the situation where F is smooth and r 0 nr 1 = 0. Let {w°, w 1 , #°(-)} be given in D(M). The solution
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{w, w', w,(-)} of (2.5)-(2.8) then is continuous from [0,oo)-» D(s$), where D(M) is topologized with the graph norm of si. That is to say,
(2.18) the mappings f-» w,, f-»— w, are continuous from [0, oo)-» W, as and w0 = w; The question is what additional regularity (if any) {w, w', w,} possesses. Let us note that because of (2.18) we have
Regularity of solutions of (2.19) with (2.17), (2.20) may be deduced from the discussion in § 4.2.1 of Chapter 4. We need to distinguish the two cases y = C and y ^ 0. (i) y = 0. On the basis of Theorem 4.4.3, we may conclude that continuously, Writing out (2.21) we have
where h e C([0, oo); H3(fl)). Equation (2.23) may be rewritten
Let us assume that Then the right-hand side of (2.24) belongs to C([0, oo); H\fl)). Upon inverting the integral operator in (2.24) using, for example, the standard Picard iteration process, we obtain
in view of (2.22). If also/,, =/i2=/22 = 0, then (see Remark 4.4.3)
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If we further assume that then
(ii) y * o. Assume that {w°, wl, &°( •)} e D(s4). Then by virtue of Theorem 4.4.4, continuously. Consequently, if (2.25) is also valid, we may conclude Suppose that
where Then (see the discussion following Theorem 4.4.4) continuously. If (2.25) also holds, we obtain that w satisfies (2.26). We note that (2.30) will be satisfied if {w°, w\ #°( •)} satisfy
and the compatibility condition involving w 1 and w"(0) in (2.30) (see Remark 4.4.5). To obtain a classical solution when y ^ 0, we assume that the data satisfy (cf. (4.4.43)-(4.4.45))
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that (2.31) is valid, and also that
We then obtain continuously. If (2.28) is also valid, then (2.29) will hold. 2.4. Strong asymptotic stability of solutions of (2.5)-(2.8). Strong asymptotic stability of solutions of (2.5)-(2.8) may not be established by the same technique that was employed in the purely elastic case of Chapter 4, since the resolvent of —^~lsi is not compact in the present situation. We may, however, establish strong asymptotic stability by employing the well-known invariance principle of LaSalle. The use of this principle in the context of boundary stabilization problems is well established (see, e.g., Slemrod [1], Dafermos [1], [2]), and we follow the approach discussed in Dafermos [2]. To be specific, let {T( t) \ t g 0} be the contraction semigroup on 3f generated by — Va^M. For any e$f, the orbit through $ is by definition 77(
for some ta e 3ft. From the definitions of s& and % it is easily seen that (2.32) is the same as
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If 01 = 0, then v = 0 and from (2.33c) we obtain u(s) = w. Then (2.33b) gives A(w + Lw) = 0; hence w + Lw = 0 since r c ^0, that is, D^w^O. Therefore, M = t) = w = 0 i f o ) = 0. Suppose that o> ?^ 0. From (2.32) and (2.12) we obtain (where ^if now denotes the complexification of the previous Sf)
and therefore
Since D"(s)^0, we conclude from (2.34b) that From (2.33c) we obtain H(S) = e~"°sw which, upon insertion into (2.35), yields But (2.36) can hold for all s g 0 only if £>"(•$) = 0 or w = 0. The former possibility implies D'(s) = 0 since D'^ = 0, which contradicts assumption (1.5). Therefore, w = 0 and then u(s) = 0, v = 0. We have proved the following result. THEOREM 2.2. Assume that T is smooth, r o nr, = 0, F 0 ^0, that the kernel D( • ) satisfies (1.5), (1.6), and that the gain matrix F is positive semidefinite on IV Then E(t)->0 as t^oo for each solution of (2.5)-(2.8). Remark 2.1. Let us note that strong energy decay holds even if the boundary conditions on Tl are conservative, that is, if the matrix F is zero, in contrast to the elastic Kirchhoff plate where energy decay occurs solely because of boundary dissipation. Remark 2.2. As in the purely elastic case considered in the last chapter, the system (2.5)-(2.8) is not strongly asymptotically stable in the space 2fifr0 = 0. since A = 0 is then an eigenvalue of ^~lsi. However, by modifying the feedback laws (1.15) as was done in § 4.2.2 of Chapter 4 (see (4.4.51)-(4.4.53)), strong asymptotic stability in $f is then restored, provided the matrix F is positive definite on a set in F of positive measure. 3. Asymptotic energy estimates. We assume that the initial data {w°, w\ #°( •)} are such that the solution of (2.5)-(2.8) is classical (see § 2.3).
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It is also assumed that (fl, r 0 ,r,) satisfy the geometric conditions (4.5.1), (4.5.2), that r0 5*0, and that the gain matrix F satisfies (4.5.3) and (4.5.4). 3.1. An energy identity. Let us introduce w* = L(w, — w), that is,
We also introduce the notation
We apply Lemma 5.1 of Chapter 4 to the function D^w + w* and integrate the resulting identity in t from 0 to T. In view of (1.7), (1.8), (1.15) we obtain
The first term in (3.2) is calculated as follows:
where
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The second term on the right-hand side of (3.3) may be written
We have
Therefore, (recall that £>(0) = 1),
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Substitution of (3.7) into (3.5) yields
Use of (3.3) and (3.8) in (3.2) allows (3.2) to be written
We next transform the fourth term on the left-hand side of (3.9) as follows:
where
We have (cf. (4.5.13))
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From (3.6)
It follows from (3.11), (3.12) that
Use of (3.10) and (3.13) in (3.9) yields
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Let us examine next the boundary integrals over F0. Write ar = a ro +a r| . Since w = dw/dv = 0 on F0, we have
Furthermore,
Therefore, the integrals over F0 in (3.14) combine to give
so that (3.14) may be written
Next, in (3.15) we consider the term
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We use Green's formula (4.3.20) with w and w replaced by Dxw + w* and w*, respectively. The resulting expression may be written
Use of (1.7) and (1.8) allows (3.17) to be rewritten (after an integration in t)
where Substitution of (3.18) into (3.16) yields
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
The second term on the right is calculated as follows:
That is,
provided that we assume
If we now insert (3.22) into (3.20) we obtain
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Using (3.6), the second term on the right-hand side of (3.24) can be written
Use of (3.25) in (3.24) gives
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We use (3.26) to replace the corresponding term in (3.15). The result is the following identity:
The identity (3.27) is the basis for the energy estimates to be derived in the next section. 3.2.
Energy estimates. We define
Then We note that
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for some constant C, where E ( t ) is given by (1.13). Indeed,
Since the form a( •; •) is strictly coercive on W, we have for some constant a2,
Estimates like (3.32) also hold for ||Vu>*(Of and ||V(m • Vw*(r))f. It follows that
We introduce the function We are going to prove that for e > 0 and sufficiently small,
where
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Let us suppose, for the moment, that (3.35) has been established. We then obtain
where From (3.33) and (3.34) we obtain If, therefore, eCg 1, we obtain from (3.36)
Equation (3.38) is valid for all initial data for which the right-hand side of (3.38) is finite. We therefore have proved, modulo (3.35), the following energy estimate. THEOREM 3.1. Assume that the kernel D(s) satisfies (1.5) and (1.6) and that the initial data {w°, w\ #°(-)} satisfy
Suppose further that T0^0, that T0,T, satisfy (4.5.1), (4.5.2), and that the gain matrix satisfies (4.5.3), (4.5.4). Then the solution of (1.1)-(1.4), (1.15) satisfies
for some w > 0. Remark 3.1. u> is given by
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where C is defined by (3.33) and where 0< e < e0 with e0 depending on elastic and geometric parameters and on the upper and lower bounds on the gain matrix. The value of e0 will be made explicit in the derivation of (3.38). We would like to deduce a stronger decay rate for E ( t ) than (3.42). If it could be shown that
for all initial data {w°, w1, #°(-)l e <%•, then a uniform exponential decay rate for E ( t ) would follow from a result of Datko [1]. However, (3.42) cannot be expected to hold for all {w°, w1, -&0(-)}e Sf without further restrictions on D( •). The reason for this lack of uniform asymptotic stability may be seen from (1.13) and (1.14) by noting that the boundary feedbacks (1.15) affect only the instantaneous kinetic and strain energies and not the memory term in (1.13). Furthermore, it is easy to see that the memory term cannot be influenced by any boundary feedback involving only the current state of the system; its asymptotic behavior is determined solely by asymptotic properties of the kernel D(s). It is therefore necessary to impose further restrictions on the kernel in order to improve the asymptotic properties of E ( t ) . In this regard we have the following result. THEOREM 3.2. Assume that the conditions of Theorem 3.1 are met. Suppose further that d° has compact support in [0, oo) and that for some m^O,
Then the solution of (!.!)-(1.4), (1.15) satisfies
If for some 17 > 0,
then for some w > 0,
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
143
Proof. The proof of (3.44)m uses the key inequality (3.35) together with a nice trick of Leugering [2] (introduced in the course of a derivation of (3.44)m in the context of a viscoelastic membrane with boundary damping). The proof is by induction on m. Assume (3.43)0, multiply (3.35) by t, and integrate the product from 0 to t to obtain
Using (3.37) and integrating by parts in the last term on the right we obtain
We rewrite the last integral on the right as follows:
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in view of (3.40), (3.43)0, and the compact support of ft0. The estimate (3.44)0 follows from (3.47) and (3.48). We now assume that (3.44) m _j holds, we multiply (3.35) by f m + 1 , and integrate from 0 to t. Proceeding as above we get
The last integral equals
Equation (3.44)m now follows from (3.49), (3.43)m, and the induction hypothesis (3.44)m_!. The proof of (3.46) is similar. We multiply (3.35) by e f t ) T ( 0 < w g i j ) and integrate from 0 to t. After integration by parts and use of (3.37) we obtain
ENERGY ESTIMATES FOR VISCOELAST1C PLATES
145
Let
The final term in the last inequality is bounded above by
so that we have
The proof of (3.46) is completed by choosing w > 0 so small that
Remark 3.2. The derivation of (3.35) and, consequently, Theorems 3.1 and 3.2, use the positive definiteness of the gain matrix F in an essential way. On the other hand, various asymptotic decay rates for the solution may be obtained without recourse to boundary feedback, i.e., based only on properties of the kernel (see Desch and Miller [1], Priiss [1]). It would be of interest to determine precisely what effect the feedback laws (1.15) have on stability, in particular, whether such boundary feedback increases the margin of stability. 3.2.1. Proof of (3.35). We have
The derivative p'(t) is calculated using (3.28) and the identity (in T) (3.27). The result is (writing t in place of T)
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We proceed to obtain an upper bound for p'(t). First, we estimate
where al is the smallest constant such that The next estimate is obtained in a similar fashion:
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
where «2 is the smallest constant such that
Use of the estimates (3.52) and (3.53) in (3.51) yields the upper bound
where
Let us estimate the last term on the right-hand side of (3.55).
Recall that b(
where G satisfies
with 0
147
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The terms in (3.59) containing derivatives of D^w + w* up to order one can be bounded above by (const.)a(Dccw + w*), while the terms containing secondorder derivatives can be estimated by (const.)a ri (D x w + w*). Therefore, we have an estimate of the form
for suitable constants /3t and /32. To estimate a(w*) in (3.60), let d"
Therefore,
from which it follows that
for a suitable constant fi}. Inserting (3.61) back into (3.60) gives
We also have
ENERGY ESTIMATES FOR VISCOELASTIC PLATES
We may now use (3.62), (3.63) in (3.57) to obtain
Substituting (3.64) for the corresponding term in (3.55) yields
where Choose 77 > 0 so small that
We then obtain from (3.65) the estimate
where
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Therefore, from (3.50) and (3.66)
where D0 = min (1 -Dx, D^). From (3.58) it may be seen that
provided y = Ji2/12 = 1, as we may assume. Consequently,
The proof of (3.35) is now completed by choosing e > 0 so small that
Chapter 7
Uniform Asymptotic Energy Estimates for Thermoelastic Plates
1. Orientation. We consider the thermoelastic plate model (1.41)- (1.45) of Chapter 2 under the conditions Assumptions (1.1) mean, respectively, that there are no heat sources within the plate, and that the heat flux through either face of the plate vanishes. We make the change t -> t\iD/ ph in the time scale and also change notation in order to bring the system to the form
where a, ft, cr, and 77 are positive constants,
The initial conditions for (1.2)-(1.5) are The functions t>, and v2 are the control variables. However, # is not a control variable since it is a measure of the temperature of the external medium along 151
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the edge of the plate and as such is not a quantity which can be expected to be at our disposal. However, it is evident that some restrictions on 1? will be required if we expect to be able to stabilize the motion of the plate through feedbacks in the moments and shear force on f. These restrictions will be discussed in the next section. 1.1. The energy functional. In deriving energy estimates, the strain energy of the plate in bending, 9b, as denned in (2.1.40) is not convenient to work with because of the term (I + /A)#AW, which renders the possibility that &b may be indefinite. Of course, one may argue that the assumptions of the model imply that |#| is sufficiently small to assure that 3Pb is positive definite (if F 0 ^0). Nevertheless, smallness of |#| is not an assumption that we wish to make in our derivation of energy estimates. Therefore, we shall introduce a stronger "energy" functional, but one which is natural for the problem (1.2)(1.5). To this end, we multiply the two equations (1.2) by w' and #, respectively, add the products, and integrate the sum over H. We obtain
Using Green's formula (4.3.20) and the boundary conditions (1.4), the first term in (1.7) may be written
The last term in (1.7) is
It follows from (1.8) and (1.9) that
In addition, from (1.5) we have
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
153
Substitution of (1.10) and (1.11) into (1.7) allows (1.7) to be rewritten
where the terms on the right are evaluated at time t and where is what we shall call the thermoelastic energy. Note that for some positive constant C. We now choose where ^,, ^2 are denned by (3.23) and (3.24) of Chapter 4. Then (see (4.3.22))
where F is a symmetric 3 x 3 matrix [/-,-] of L^FJ functions which is assumed to be (at least) positive semidefinite on F,. The last term in (1.12) is then nonnegative and represents mechanical dissipation in the system. In order to obtain a uniform asymptotic decay rate for (1.13), we shall have to require that the system also be thermally dissipative, that is,
Equation (1.16) is a restriction on the thermal parameters and/or the temperature of the surrounding medium. It will be satisfied if, for example, A = 0 (insulated edge condition); or if d sgn #^ k\&\ on F, where k satisfies
The latter inequality will be satisfied for \k- 1|A sufficiently small, depending on 0-/17 and on the diameter of ft. Remark 1.1. The issue of asymptotic stability of solutions of the equations of linear thermoelasticity was considered by Dafermos in [2], [3], where it is demonstrated that thermal dissipation alone is sufficient to induce strong asymptotic stability of solutions. That work suggests that the system (1.2)-(1.5) is strongly stable even in the absence of dissipative boundary mechanisms and, indeed, such a result will be established in the next section (in the case F0 ^ 0). In later work, Slemrod [2] studied the question of asymptotic stability
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in the context of one-dimensional, nonlinear thermoelasticity. His results, when specialized to the linear case, imply that the thermoelastic energy decays at a uniform rate. It seems unlikely, however, that such a result is valid for higher dimensional linear thermoelastic systems, including the one considered here, unless additional dissipative mechanisms (such as boundary damping) are introduced. 2. Existence, uniqueness, regularity, and strong stability. 2.1. Well-posedness of the problem. We consider the system (1.2)-(1.6) with vt, v2 given by the feedback laws (1.14), and we assume To simplify the discussion a bit we set # = 0 in (1.5). It is our intention to put the system (1.2)-(1.5), (1.14) into an abstract form to which semigroup theory may be applied, along the lines of the existence theory developed in Chapter 4 for the elastic Kirchhoff plate model. Actually, existence of solutions may be proved rather directly by using a Galerkin approximation scheme. We prefer, however, a semigroup approach in order to obtain solutions that have at the onset better regularity properties than those that result from the Galerkin method. The semigroup approach also allows us to obtain additional regularity properties on the basis of the regularity discussion of § 4.4.2.1. To motivate the abstract formulation of the system (1.2)-(1.5), (1.14), suppose that {w, &} is a regular solution of this problem. The pair {w, &} is then a solution of the variational equation
where the bilinear form a( •; •) is defined by (4.4.4). We set
and we identify H with its dual H'. We then have the continuous and dense embeddings W<=• Tc 3ffc T c W. We introduce the following bilinear forms:
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
155
The form c( • ; • ) is continuous, symmetric, and coercive on Y, while b( • ; • ) is continuous on W and satisfies
for some A^ > 0, where b({w, #}) = b({w, #}; {w, #}). We define operators Cz&(Y,Y'), Be£(W,W), A0e£(W,W), A^ %(W, W) by
The operator C (respectively, A0) is an isomorphism of Y onto Y' (respectively, W onto W) and is the canonical isomorphism if we introduce c( •; • ) (respectively, a( • ; • ) ) as a scalar product on Y (respectively, on W), as we now do. (The statement regarding A0 requires r 0 7 £ 0.) The variational equation (2.2) may be written in terms of the above operators as an equation in W as follows: Let us introduce the notation and an operator Pe£(W, W) by
Then (2.6) may be written as the system
or, in matrix form,
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We set
Then <€ is the canonical isomorphism of W x V onto W x °V\ and A e £(WxW, W'xW). We introduce the set
and we define
Then9T1.s£:D(.!rf)-» W x T is an unbounded operator in Wx°V=WxVxH. Remark 2.1. From (2.8) it is seen that 3> = {
It may then be proved that (cf. Remark 4.4.1)
THEOREM 2.1. -•# 1M is the infinitesimal generator of a Co-semigroup of contractions on WxVxH. Proof. From (2.9) it follows that D(M) is dense in W x V x H. Let * e D(st). Then
We next show that
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
157
Since <£ is an isomorphism of WxT onto W'xY', condition (2.10) is equivalent to
Consider the equation
that is,
The system (2.11) is the same as
Use of the first equation of (2.12) in the second equation yields
Since for a positive constant k^ we have
(2.13) has a unique solution {<£,,
that is, {(p0, i p t , ip2} e D(M).
Remark 2.2. Assume that {w°, w\ -&°}e D(sd). Then the map f-» {w((), w'(0, *(')}: [0, <») -» D(J^) is continuous, where D(J^) is equipped with the norm \\^^sd{tp0,
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Remark 2.3. When -y = 0 in (1.2) and (1.4), then C = I, the injection of 5if onto itself, and %> is the canonical isomorphism of W x "X onto W x 3V. In this case, -(&~ls& generates a Co-semigroup of contractions on WxHxH, where
2.2. Regularity of solutions. Let {w°, w1, #°}e D($$). Then the solution satisfies { w ( t ) , w'(t), d(t)}e Wx Wx U for t^O,
Let us examine the property (2.16). Let {w, #}e Wx U. Then
so that (2.16) may be written
where /j e C([0, oo); V) fl L°°(0, oo; V), /2 G C([0, oo); H) H L°°(0, oo; H). (In fact, /2 = -fid' and/, = -C0w", where C 0 e^(V, V) is defined by (C0v, v) = ( u , v ) + y(Vv,Vv), for all v,veV. If y = 0 then /,e C([0,oo); H)H L°°(0, oo; //).) If we set w = 0 in (2.17), we obtain
Equation (2.18) means that d(t) is the solution of
The right-hand side of (2.19) is in C([0, oo); H) D L°°(0, oo; H). It follows from elliptic theory that
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
159
Next, we set # = 0 in (2.17) and obtain
The last equation may be written
where g, =/,-aA#e C([0, oo); V) D L°°(0, oo; V). Equation (2.22) is essentially the same variational equation that was studied in § 4.4.2.1, and we can draw some immediate conclusions based on that earlier discussion.
In addition, if/,, =/,2 =/22 = 0 in the matrix F, then Inclusions (2.23), (2.15), and (2.21) show that -^ si has compact resolvent when y = 0. (ii) Suppose y ^ 0. Then whenever the data {w°, w\ #°}e D(sd). If, in addition, the data satisfy
we may conclude that In (2.27), {w"(0), tf'(O)} is calculated from (2.6): Hypotheses (2.25) and (2.26) guarantee that so that {w"(0), #'(0)} e W x U and the right-hand side of (2.27) has a meaning.
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One may also establish the existence of classical solutions of (1.2)-(1.6) under suitable conditions on {w°, w 1 , #°} (cf. (4.43)-(4.45) of Chapter 4). 2.3. Strong asymptotic stability. Since the above regularity properties of the solution imply that the resolvent of -<£~ l sd is compact, strong asymptotic stability is proved once it is established that C€~1M has no eigenvalues on the imaginary axis. This can be expected to be the case only if F0 ^ 0, which we assume. Suppose that A = ia> (&> 6 R) is such that
for some O = [
If w = 0 it is easy to see that $ = 0 (since F 0 ^ 0). Suppose that w ^ 0. From (2.30) we have
If we take the inner product of (2.30) with {
Note that
The first term on the right-hand side of (2.33) is purely imaginary while the rest is real and nonnegative. It therefore follows from (2.32) upon taking imaginary parts that
Consequently,
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
161
Then, from (2.31) we obtain
Since o> ^ 0, (2.35) signifies that
so obviously
Uniform asymptotic energy estimates.
3.1. An energy identity. Let {w, •&} be a solution of (1.2)-(1.5). We shall assume that this solution is sufficiently smooth to justify the computations that follow. In particular, this will require that w have at least H7/2+s(tl) regularity and # at least H3/2+s(fl) regularity in the spatial variables for some 5>0. From the first equation in (1.2),
We have (cf. § 4.5.2)
where
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where, for any measurable set A c Y,
(We note that a ro (w) = -Jro m • v(kw)2 dT if m • v^O on T0. For reasons of notation only, we assume m • v ^ 0 on F, for the remainder of this section. However, this geometric assumption will be essential in the derivation of a priori estimates given in the next section.) We also have
Substitution of (3.2)-(3.4) into (3.1) yields
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
Next, we use
where From (3.5), (3.6) we obtain
We next employ the identity
163
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CHAPTER ?
Multiply (3.8) by @/2cr and add the product to (3.7) to obtain
where
3.2. A priori estimates. In order to obtain uniform asymptotic energy estimates we shall need the following geometric assumptions (stronger than those of the previous chapters):
We also suppose that
and that the gain matrix F is uniformly positive definite on F,. Because of (3.10), we may then write
where g0 and G0 are constants. Then b(u; v) has the form
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
165
and satisfies
Finally, we shall require that the system (1.2)-(1.5) be strongly thermally dissipative, that is,
for some positive constant fc (cf. (1.16)). Under the above assumptions, a uniform decay rate will be obtained for regular solutions of (1.2)-(1.5). The derivation of a priori estimates is based on the identity (3.9). We introduce
Then
Using (3.11) and Poincare's inequality we have
for an appropriate constant C. For e > 0, define
It will be proved that for e > 0 sufficiently small,
where
Uniform energy decay then follows from (3.16), (3.18) by an argument that has been made repeatedly in the previous chapters.
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To prove (3.18), we first estimate p'(t), which is calculated by differentiating (3.9) in T:
where the terms on the right are evaluated at t. We proceed to estimate various of these terms. Let 8 > 0 be arbitrary. We have
The term b(w';m-Vw) is estimated along the lines of (5.21)-(5.24) of Chapter 4: for any £> 0,
The next to last term on the right-hand side of (3.19) may be estimated in the following way, keeping in mind assumption (3.10):
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
167
The last term on the right-hand side of (3.19) is bounded from above with the aid of assumption (3.14), and one obtains
for some constant Cl. If the inequalities (3.20)-(3.23) are substituted for the corresponding terms in (3.19), an estimate for p ' ( t ) is obtained that has the following structure:
We choose 5 = ^ in (3.24) and obtain an estimate of the form for some constant C^. The proof of (3.18) may now be completed as follows. We have From (1.12), (1.15), and (3.14),
If (3.25) and (3.27) are inserted into (3.26), the result is
76 8
ch apt e r 7
provided y = h2/l2§ 1. The estimate (3.18) follows by choosing e > 0 so that
As a consequence of (3.18), we obtain by the usual calculations
where
with C denned by (3.16). Remark 3.1. The requirement that the constant y > 0 in (1.2) was essential to the above proof. 3.3. The case of small diffusion. An interesting connection to the viscoelastic situation studied in Chapter 6 arises if it is supposed that the diffusion term rjAtf in (1.2b) is so small that it may be ignored. (Of course, the boundary condition (1.5) should then also be dropped since it arises as a result of diffusion.) Equation (1.2b) is then an ordinary differential equation for # in which {x, y} enter as parameters. Let us assume that the evolution of {w, -&} starts at t = -co with w(-co) = 0, #(-oo) = 0. Upon integrating (1.2b) we obtain
Define
Then
so that, upon substitution of (3.29) into (1.2a), we obtain
ENERGY ESTIMATES FOR THERMOELASTIC PLATES
169
Equation (3.32) may, in turn, be rewritten as
We note that so that (3.33) has exactly the same structure as the viscoelastic plate equation (4.1.1). By a change of time scale in (3.33), we may assume that D(0) = 1. Let us rewrite the boundary conditions (1.4). The first of these is
That is,
Similarly, the second boundary condition on (1.4) may be written
The boundary conditions (3.34), (3.35) have exactly the form of (1.3) of Chapter 6. Furthermore, because of the structure of the kernel (3.30), the system (3.33)-(3.35) will be uniformly asymptotically stable in $? (defined in Chapter 6, § 3.2) even if £, = v2 = 0 (see the discussion following Theorem 3.1 in Chapter 6, § 3.2). In terms of the original control functions vl,v2, this means that one may select for the control laws
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Index
Airy stress function, 8, 9, 19
Kinetic energy, 13-16, 18, 26, 27, 40, 65, 120 in bending, 2, 14-16, 22, 111, 153 in stretching, 14 Kirchhoff equation, 1, 68, 69, 89, 90 hypothesis, 21, 22 Korn's lemma, 29, 39
B o u ndary control s , 7 - 9
Coefficient of thermal expansion, 23 Control laws, 169 Control variable, 3, 151 Decay rate, 3-5, 7-10, 54, 55, 61, 90, 142, 145, 153, 154, 165 Dirichlet data, 34-36 Elastic plates, 8, 10 Elastic-viscoelastic correspondence principle, 21
Lame coefficients, 21 Laplace transform, 20, 21 Lasalle's invariant principle, 129 Linear-quadratic-regulator (LQR) framework, 7 Middle surface, 2, 13, 14, 16 Mild solution, 32
Exact boundary controllability, 7, 10, 40 Exponential decay, 9, 55, 84 Feedback controls, 1, 7, 9, 27, 120 law, 3,9-11,39,40, 79, 90, 111, 130, 145, 154 Finite energy solution 28, 32, 38, 79 Flexural rigidity, 2 modulus of, 11 viscoelastic, 21
Neumann boundary condition, 4, 6, 10, 11, 54, 57, 102 Nonlinear-feedback law, 99 w-limit set, 109, 129 Orbit, 129 Plane elasticity, 15, 35, 62 Plate models Kirchhoff, 8-10,14,15,20,21,24,65,130,154 Mindlin-Timoshenko (M-T), 8, 15, 16 thermoelastic, 8, 22, 24, 151 viscoelastic, 8, 9, 20, 169 von Karman, 4, 8, 9, 17, 20, 24 Poisson's ratio, 2, 13, 21, 29 Principle of virtual work, 15, 24
Green's formula, 26-29, 33, 37, 42, 46, 67-69, 73, 83, 92, 99, 111, 115, 120, 136, 152 Hilbert uniqueness method, 8 Holder's inequality, 107 Homogeneous material, 10, 13, 101 Isotropic material, 10, 13, 20 175
176
INDEX
Regularity of solutions, 5, 6, 32, 36, 72, 75,113, 126, 127, 158, 165 Relaxation function, 20, 21 Relaxation kernel, 9, 10
Systems distributed parameter, 1, 11 Kirchhofl, 61, 69, 88, 99, 111, 124 Mindlin-Timoshenko (M-T), 41, 61-63, 65 von Karman, 19, 110, 113
Shear correction coefficient, 16 Shear modulus, 8, 9, 16, 61 Sobolev Imbedding Theorem, 55, 106 Stability asymptotic, 9, 37, 40, 41, 61, 62, 78, 79, 90, 129, 130, 141, 153, 160, 169 strong, 3, 9, 10, 27, 90, 153, 154 uniform, 1, 3, 6, 7, 10, 11, 15,99 Strain energy, 13, 14-18, 22, 26, 120, 142, 152 in bending, 2, 14-16, 22, 111, 152, 153 in stretching, 14
Thermal conductivity, 24 Thermoelastic energy, 10, 153, 154 Total energy, 2, 8, 26, 64, 84, 111, 120, 122 Transverse shear, 8, 9, 61, 65 Viscoelastic energy, 9, 120 Viscoelastic Poisson's ratio, 21, 119 Wave equations, 4, 6, 7, 10, 11, 40, 102, 109 Young's modulus, 13
