Appl Math Optim 19:243-290 (1989)
Applied Mathematics and Optimization © 1989 Springer-Verlag New York Inc.
Exact Controllability of the Wave Equation with Neumann Boundary Control* I. Lasiecka and R. Triggiani Department of Applied Mathematics, Thornton Hall, University of Virginia, Charlottesville, VA 22903, USA
Abstract. We consider the wave equation defined on a smooth bounded domain l~ c R n with boundary F = Fou F~, with Fo possibly empty and F~ nonempty and relatively open in F. The control action is exercised in the Neumann boundary conditions only on FI, while homogeneous boundary conditions of Dirichlet type are imposed on the complementary part F0. We study by a direct method (i.e., without passing through "uniform stabilization") the problem of exact controllability on some finite time interval [0, T] for initial data on some preassigned space Z = Z~ x Z2 based on l~ and with control functions in some preassigned space Vx, based on FI and [0, T]. We consider several choices of pairs [Z, Vx,] of spaces, and others may be likewise studied by similar methods. Our main results are exact controllability results in the following cases: (i) Z = nro(f~) x L 2 ( ~ ) . and Vx, = L2(]~1); (ii) Z = 1 r L2(I)) x [Hro(f~)] and V~, = [Hi(0, T; L2(F~))] ', both under suitable geometrical conditions on the triplet {1-1,Fo, F~} expressed in terms of a general vector field; (iii) Z = L2(Sq)x [H~(f~)] ' in the Neumann case Fo = • in the absence of geometrical conditions on f~, but with a special class V~ of controls, larger than L2(E). The key technical issues are, in all cases, lower bounds
* This paper was presented by the second named author at the IFIP WG 7.2 Conference on Boundary Control and Boundary Variations held at the Department de Math6matiques, Universite de Nice, France, June 10-13, 1986 (a preliminary version will appear as Lectures Notes in Control Sciences, edited by J. P. Zolezio [T4]); at the International Conference on Control of Distributed Parameter Systems, held at Vorau, Austria, July 6-12, 1986; at the Second Workshop on Control of Systems Governed by Partial Differential Equations, held at Val David, Quebec, Canada, October 5-9, 1986; and at the 26th IEEE Conference on Decision and Control, held at Los Angeles, CA, December 9-11, 1987 [LT6]. This research was partially supported by the National Science Foundation under Grant No. NSF 8301668 and by the Air Force Office of Scientific Research under Grant No. AFOSR-84-0365.
244
I. Lasieckaand R. Triggiani on the L2(~)-norm of appropriate traces of the solution to the corresponding homogeneous problem. These are obtained by multiplier techniques.
1.
Introduction, Literature, Statement of Main Results
1.1. Statement of Problem and Literature Let fl be an open bounded domain in R ~ (n - 2 ) with sufficiently smooth boundary 0fl = F. We assume that F consists of two parts: Fo and F1, Fou F1 = F, with Fo possibly empty and F1 nonempty and relatively open in F. We consider the exact controllability problem for the solution y(x, t) of the wave equation
'y,=Ay
in
Q = f l x ( 0 , T),
y(x,O)=y°(x), yt(x,O)=y~(x) y=0
in
Oy
-- = v
in
(1.1a) in
1~,
(1.1b)
Eo=Fo×(0, T),
(1.1c)
E~ = F~ x (0, T),
(1.1d)
where A is the Laplacian acting on the n-dimensional space variable x and v is the unit normal of F pointing toward the exterior of [l. We likewise set ~-F x (0, T). Qualitatively this means: given {[l, Fo, F1} we ask whether there is some Tm> 0 (depending on the geometry of the triplet) such that if T > Tin, the following steering property of (1.1) holds true: for all initial data yO, y~ in some preassigned space Z = Z~ x Z2 based on [~, there exists a suitable control function v on some preassigned space V~1 based on F1 and [0, T], whose corresponding solution of (1.1) satisfies y ( . , T ) = y , ( . , T ) - O . We then say that the dynamics (1.1) is exactly controllable on the space Z over the interval [0, T], by means of control functions v ~ V~1. We consider various choices of pairs [Z, V~,] of spaces. In the sequel we use the notation y ( T ) and yt(T) for y ( . , T) and y~(., T), respectively. Exact controllability problems are of course classical ones, and have received considerable attention in the literature. We therefore describe, at first qualitatively, the contribution of this paper in the context of studies on exact controllability for second-order hyperbolic equations of the past several years. In doing so, it seems advisable to examine separately, and contrast, two cases: when the control funcUon acts m the Dmchlet boundary condition and when it acts m the Neumann boundary condition. Moreover, it is necessary to weave this work and methods on exact controllability and uniform stabilization of the wave equation with two other intimately connected problems: .
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¢
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(i) Recent progress on the maximal regularity issue of second-order hyperbolic solutions, and the techniques responsible thereof. (ii) The corresponding optimal quadratic cost problem (and resulting Riccati equations).
Exact Controllabilityof the Wave Equation with Neumann Boundary Control
245
Having selected a class of boundary controls, say L2(E), it is in any case most desirable to achieve exact controllability in the space of maximal regularity of the hyperbolic solutions. This space then naturally identifies the space for penalization in formulating quadratic cost problems. Then exact controllability, once achieved, automatically guarantees "the finite cost condition" in the regulator problem [LT5], [FLT]. With this goal in mind, there are at present notable differences regarding exact controllability results available for the wave equation, according to whether the control action, say in L2(E), is exercised in the Dirichlet or in the Neumann boundary conditions (perhaps only a portion of the boundary). Dirichlet Boundary Control. Here, results obtained prior to the mid-eighties referred to smooth initial data: H2(I~)× H~(f~) or C~(f~)x C~(I~) in the case of general geometries, see [R1]-[R3], [L1], [L2], and ILl0], [Lll]. It was later discovered [L4], [LT1], [LT4], [LLT] that the space of maximal regularity of second-order hyperbolic solutions with L2(E)-Dirichlet controls is, in fact, L2(f~) x H-l(f~). This, in turn, is related to a trace regularity result: an upper bound estimate of the L2(~)-norm of the normal derivative of the solution of the corresponding homogeneous problem in terms of the H~(I~) × L2(f~)-norm of the initial data, for all 0 < T < ~. As a consequence, it was only recently that the desirable goal was achieved of establishing exact controllability in the space L2(I)) x H-l(f~) of maximal regularity. First, by Lasiecka and Triggiani [LT2], as a corollary of the more demanding uniform stabilization in L2(f~)x H-~(f~) through an explicit feedback control, which then requires geometrical conditions on f~ (e.g., strict convexity); then by Lions [L5]-[L8] by his newly developed {F, F', A}-method, a direct approach which manages to dispense altogether with geometrical conditions on f~ (except for smoothness of F), if the control is exercised on the entire boundary F, by relying on the corresponding lower bound estimate of the normal derivative valid for T sufficiently large (this inequality was explicitly obtained by Ho [H1]; and it is essentially contained, in a less transparent form, as a crucial ingredient in the proof of the uniform stabilization result previously obtained by Lasiecka and Triggiani [LT2]); and by Triggiani [T2] by another direct approach ("ontoness" or the input-solution operator) which extends the results of [L5]-[ L9] and [ H 1] to more general triplets {l~, Fo, F1} in the case where the control is applied only on the Subportion F1 of the boundary F = Fo u F1. In all these works the same class of multipliers, first used in the maximal regularity theory [L4], [LLT], plays an essential role, and unifies them all: it provides an identity (see (2.18)), which is then used to obtain either the upper bound required for the trace theory result, or else the lower bound required for uniform stabilization and exact controllability. Neumann Boundary Control. Here, sharp results on regularity of the solutions with, say, L2(E)-controls are just emerging, see [LT3], since the multipliers and the methods used in maximal regularity theory for the Dirichlet case are insufficient now. While in the case of dim ~ = 1 the space of maximal regularity is Hl(f~)x L2(f~), the results in [LT3] show that for dim l~->2 the space of
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I. Lasieckaand R. Triggiani
regularity is dependent on the geometry of l~ and is at any rate strictly larger than Ha(O) x L2(fl) for, say, Fo = Q, yO= y~ = 0. Nevertheless, the space H~o(f~) x L2(l-l) is the only space where exact controllability results have been obtained with L2(Ea)-controls prior to now: by Graham and Russell [GR] in the special case of a sphere f~ by eigenfunction expansion and moment problem techniques; by Chen [C1], [C2], Lagnese [L3], Lasiecka and Triggiani [LT2, Section 4], and Triggiani IT2] in the case of general smooth l-I subject to geometrical conditions, as a corollary of uniform stabilization; by Littman [L10], [L11] by extending the idea of Hughen's principle, with no guarantee however that the Neumann control is in L2(~). By contrast, this paper studies, through a direct approach, exact controllability of problem (1.1) for different pairs {Z, Vx~}of functions spaces for the solution and the control: not only Z = H~o(fl) × L2(~-~), V:~1 = L2(]~l), as in the literature a r quoted above, but also Z = L2(12) x [Hro(12)], Vx, = [Ha(0, T; L2(Fa))]'; and, in the purely Neumann case, Fo = Q, Z = L2(E) and with a special structure. While in the first two cases, geometrical conditions on the triplet {f~, Fo, Fa} must be imposed, these are altogether dispensed with in the third case (dealing with the purely Neumann problem). The other work, which considers, likewise by a direct approach, exact controllability for (1.1) on the spaces {Z, Vz} listed above, is Lions [LS], where Fo = Q is taken throughout. The general approach in [L8] and the one in this paper are different, however. Lions uses his (F, F', A)-method. Our paper is based instead on the ontoness approach of the operator ~fT (see (3.7)) with a preassigned target space in the style of IT1]. This is crucially based on a priori partial differential equation inequalities which in turn characterize the equivalent statement that LP* has continuous inverse. These inequalities--once achieved--establish stronger versions of the Holmgren-John uniqueness theorem. Unlike the case of the wave equation with Dirichlet boundary control where Lions's space F = H~(I~) x LE(~q)[H1], the spaces F, F' are still elusive in the Neumann problem (1.1), though containment relations such as F c Ha(O)x L2(~'~) for T large may be given. Our main results are Theorems 1.1-1.4 listed below in Section 1.2. Our Theorems 1.1 and 1.2 apply to more general triplets {f~, F0, FI} than those in [LS, Theorem 5.2] which are described by general vector fields as in Section 1.2. Once specialized to radial vector fields and to Fo = Q, they then reduce to the geometrical conditions in [L8] which amount to fl being star shaped with respect to some point in R". Our Theorem 1.3 does not have a counterpart in [L8]. Finally, our Theorem 1.4 provides exact controllability of L2(~'~) x [HI(~-~)] ' in the absence of geometrical conditions on f~ with a class of Neumann boundary controls (Vlc [H~(0, T; LE(F(x°)))] ' in the structure class given by (1.15) below) which is strictly smaller than the one considered in [L8, Theorem 5.1], which takes v~c H-a(0, T; L2(F(x°))) instead. Moreover, the two approaches are different. We also note, in this respect, that once exact controllability is established, then an elementary argument provides explicitly the minimal norm steering control (see the abstract operator argument in Appendix B and its specialization there to the wave equation with Neumann boundary control). This is precisely the control used by Lions [L8]. (See also [T1, Appendix B] for the Dirichlet case.)
Exact Controllabilityof the Wave Equation with Neumann BoundaryControl
247
Finally, in comparing our exact controllability results here with those in [L3], [LT2, Section 4], and IT2], in the only common case where the control space is L2(]~1) and the space of initial data is H~o(12) x L2(~), we may note that the final statements are very similar. A common crucial technical tool in all these works, as well as in this paper, is represented by a special class of multipliers which either appear indirectly, in a Lyapunov-type functional to be differentiated in time [L3], [LT2, Section 4], or else are applied directly to the equation IT2]. However, the multiplier technique in the uniform stabilization problem [L3], [LT2], [T2] is applied to the feedback control system; in the direct approach to exact controllability of this paper, to the corresponding homogeneous system. This gives rise to some differences. In both approaches, mild geometrical conditions are required on the triplet {l~, Fo, F~}. The proofs on uniform stabilization [L3], [LT2, Section 4], [T2] require explicitly the condition Fort F1 = Q, a consequence of the assumption h. v -> constant Y > 0 on F1, combined with the needed smoothness of F. The proofs in this paper require only h. v - 0 on F1, i.e., the constant may be taken equal to zero and thus apply in principle also to the important and delicate case F o n F l ~ ~ , subject to the comments made in Remark 1.4.
1.2. Exact Controllability on H~o(f~ ) x L2(~~) with Controls v~ L2(0, T; L2(F1)) -- L2(£1) and M(yO)=0 /f F o = Q We first consider v ~ L2(£1) in (1.1d) and study the corresponding exact controllability problem on the space H1o(12) x L2(l)), where
H~-o(f~)={f~Hk(f~):f]ro=O},
k= 1,2,3,...,
even though recent studies on regularity theory for problem (1.1) with, say, Fo = •, i.e., r 1 = F and yO = yl = 0, show that the map v -~ [y, y,] is not continuous L2(£) -~ C([0, T]; H i ( o ) x L2(I~)), 0 < T < oo. Indeed,.for v c L2(£) and yO = y~ = 0, the solution [y(T), y,(T)] of (1.1) lies in a space strictly larger than H 1 ( ~ ) x L2(O), which appears to depend on the domain f~ [LT3] (example: H2/3(l~)x H-~/a(f~) for lq a sphere; H3/4-~(1~) x H-l/4-~(f~) for ~ a parallelepiped [LT1]). Nevertheless, the space Hlo(l))x L2(I)) is of physical interest ("energy space") and this justifies our study of exact controllability here. As a result, even if we find, say for Fo# Q, that any preassigned initial pair (yO, y l ) ~ Hlo(f~)x L2(~~) can be steered to rest, y ( T ) = Yt(T)= 0 for all T > some (universal) time 7",,> 0 by some suitable control v ~ L2(0, T; L2(F1)), there is no guarantee that during the transfer 0 < t < T, the solution [y(., t), y,(., t)] remains in H~o(l~ ) x L2(f~). Our main results are as follows; in particular, in contrast with the Dirichlet case [H1], [L4]-[L6], IT1], they require geometrical conditions even when F1 = F. Let the triplet {11, Fo, F1} possess a vector field h(x) = [ h i ( x ) , . . . , h,(x)] c C2(1]) such that:
(H.1)
{hh:V<-O on ro, v___0 on F1,
v = outward unit normal.
(1.2)
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I. Lasieckaand R. Triggiani
(O.2) n H(x)v(x)" v(x) d~
>-p
f. Iv(x)l ° dn OXl '
for some constant p > 0 ,
Vv ~ [L2(f~)]",
(1.3)
" ' Ox.
H(x)
"
0h°
0h.
c9Xl '
' OXn
(1.4)
•
(A sufficient checkable condition for (1.3) to hold is that H ( x ) + H * ( x ) uniformly positive definite in ~.)
be
(H.3) (1.5)
p > 2GhCp,
where 4Gh =--max f~
IV(div h)l,
(1.6) Vtp ~ H~o(fl ) defined above,
0 < Cp = Poincar6 constant.
(1.7)
Remark 1.1. Assumptions (H.1)-(H.3) apply, in particular, to (suitably smooth) triplets {f~, Fo, F1} which are "star-complemented-star-shaped" [C1], [R1]. This means that there exists a point Xo~ R ~ such that
I
(x-Xo)" v-< 0
[ ( x - X o ) " v >-0
on Fo on F1
(Fo is star complemented with respect to Xo),
(1.8a)
(F1 is star shaped with respect to Xo)
(1.8b)
and we can then take h ( x ) = x - Xo (radial field) in the statement of assumptions (H.1)-(H.3). Indeed, in this case we have H ( x ) =- identity matrix,
div h - n = dim l~,
Gh = O,
and assumptions (H.2) and (H.3) are automatically satisfied with p = 1.
Remark 1.2.
It is possible to construct smooth domains ~ (say, in the plane R2), such that hypotheses (H.1)-(H.3) for a corresponding triplet {fL Fo, F1} are satisfied by a suitable smooth vector field h ( x ) , while condition (1.8) fails for any radialfield X - X o , Xo~ R " (n =2). More precisely, for such l~, use of radial vector fields yields an active portion F~ of the boundary F strictly larger than the active portion F1 obtained via a suitable (nonradial) vector field h ( x ) satisfying (H.1)-(H.3). See [T1].
Exact Controllability of the Wave Equation with Neumann Boundary Control
249
Theorem 1.1.
Let either Fo ~ O or else Fo = 0 . Let the triplet {fl, Fo, F1} satisfy hypotheses (H.1) = (1.2), (H.2) = (1.3), and (H.3) = (1.5). Then there exists Tm> 0 (to be specified below) such that if T > Tm then, for any (yO, yl) ~ H~o(l~ ) x L2(fl), subject to the further requirements that M(y°)=-~ay°dfl=O and M(yl)=_ ~a yl dfl = 0 if Fo = O , there is a suitable v e L2(O, T; L2(F1)) such that the corresponding solution of problem (1.1) satisfies y ( T ) = y~( T) = O. For Tm we may take T~ _2(DhCp+ Mh) ÷ Cp,
(1.9)
p-20~
Mh-=m~x[h];
2Oh--m_axldivh[;, Gh andCpasin (1.6), (1.7).
(1.10)
In the main statements below, we dispense altogether with the somewhat unnatural assumption (H.3). This is possible by exploiting the following uniqueness property 1 of the set F~ over the time interval [0, Tu], 0 < Tu < oo, which depends on the triplet {f~, Fo, F1} of the corresponding homogeneous problem. It is expressed by the following implication:
{
~ott= A¢
on
~olz=0
in
E=Fx(0,
O~o = 0 av Iz,
in
E1=FIx(O,T),
T>Tu
implies
Q = l) × (0, T), T),
(1.11)
~o----0inQ.
Remark 1.3.
That the above property (1.11) holds true is a rather straightforward consequence of the Holmgren theorem, as presented by H6rmander [H2, Theorem 5.33, p. 129]. (In a previous exposition [T4] of this paper, we took this property (1.11) as an assumption.) The Neumann problem F1 = F, i.e., Fo = O (empty), is the classical Holmgren-John theorem [J1], [R1]. Indeed, this uniqueness result holds true in the general constant coefficient case. In the general variable coefficient case, if instead Fo # 0 , we understand that ongoing work by Littman [L12] should provide the required nontrivial extension of this uniqueness property. The next result dispenses with assumption (H.3) by ultimately using property (1.11). Consequently, it provides a different T,, ; moreover, it avoids the assumption ~ / ( y l ) = 0 when Fo = Q . Theorem 1.2.
Let either Fo ~ Q or else Fo = @. Let the triplet {1~, Fo, F1} satisfy assumptions (H.1)= (1.2) and (H.2)= (1.3). Then there exists T " > 0 (see below) such that if T > T " then, for any (yO, yl) e H~o(~) x L2(I't), subject to the further
1This corresponds, in modem control theoretic terminology, to an "observability" property (as "distinct from" continuous "observability" [R1]).
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1. Lasiecka and R. Triggiani
requirement that d,t(y °)=-say°d12=O /f F0---~, there is a suitable v L2(0, T; L2(F1)) such that the corresponding solution of problem (1.1) satisfies y( T) = y t ( T ) = 0. For T " we can take the following values: for For ~ ,
t
)
T~>max 2 CpDh + Mh+Cp, r~ , P
(1.12)
for Fo = ~,
1.3. Exact Controllability on L2(12)x [H~,o(12)]' with Controls v ~ [HI(0, T; U(r,))]' Let [Hi(0, T; LZ(F1))] ' be the dual space to the space Hi(0, T; L2(F0) with respect to the H°(O,T;Le(FO)-topology. We have HI(O,T;L2(FO)~ Hi(0, T; L2(F0) and hence [Hi(0, T; L2(F~))]'~ H-l(0, T; L2(F0). For v in the class of [Hi(0, T; LZ(F0)]'-controls we have: Theorem 1.3. Let either Fo ~ ~ or else F0 = ~ . Let the triplet {12, Fo, F1} satisfy assumptions (H.1) = (1.2) and (H.2) = (1.3). Then there exists T " > 0 (see below) 1 r such that if T > T " then, for any (yO, yl) ~ L2(12) x [Hro(12)], there is a suitable v ~ [Hi(0, T; L2(F1))] ' such that the corresponding solution ofproblem (1.1) satisfies y ( T ) = y t ( T ) = O . For T " we can take the same T " stated in Theorem 1.2, see (1.12) and (1.13)for Fo~ ~ and F o = Q , respectively. 1.4. Exact Controllability on Lz(12)x [H1(12)] ' in the Neumann Case (Fo = O) in the Absence of Geometrical Conditions on 12 with a Special Class of Controls v The preceding results, in particular Theorems 1.2 and 1.3, yield exact controllability for problem (1.1) with Fo = O (Neumann case) on the spaces H1(12) x L2(12) and L2(12) x [H~(fl)] ', respectively, subject at least to the geometrical conditions (H.1) and (H.2) on 12. It turns out, however, that the same analysis which culminates with these results permits us to show--modulo modifications that introduce no essential additional difficulties~an exact controllability result on L2(~'~) × [ H 1(12)]' for the Neumann problem (1.1), i.e., Fo = ~ , with no requirement of geometrical conditions on 12. Elimination of the geometrical conditions does not come without a price. The price consists in the class of admissible controls v in (1.1d) with Fo = ~ , which is considerably larger than L2(~) and, moreover, must have a special structure, as described below. Given 12, we fix a point Xo in R ~ and set (with v(x) the unit outward normal at x e F) F(x °) = {x ~ F: (x - Xo)" v(x) >-0},
(1.14a)
F . ( x o) = F\F(xO).
(1.14b)
We then define the following class of controls v in (1.1d) with Fo = ~ : ~v~ ~ [Hi(0, T; L2(F(x°)))] ' v = [vz a L2(0, T; [Hl(r,(x°))]').
dual to Hi(0, T; L2(F(x°))),
(1.15)
Exact Controllability of the Wave Equation with Neumann Boundary Control
251
We then have:
Theorem 1.4. Let F 0 = O . There exists Tin>0 (see below) such that if T > 7",, then, for any (y0, y~) E L2(fl) x [HI([~)] ', there exists a suitable control v in the class described by (1.15) such that the corresponding solution of problem (1.1) with Fo = Q satisfies y ( T ) = y,( T) = O. For Tm we can take Tm = max{ T,, Cp + 2 m~xlx - xol},
(1.16)
where T, is the time defined by the uniqueness property (1.11) with Fo = Q, see Remark 1.3. Remark 1.4 (on the triplet {fl, Fo, F1}). The proofs of Section 2 below require the existence of a dense set of initial data of the homogeneous problem (2.12), whose corresponding solutions ~ possess the regularity required to justify the calculations based on the multipliers below, in particular the left-hand side of identity (2.18). In the case Fon F1 = Q, this requirement is fulfilled provided F is sufficiently smooth. This issue, however, is far more delicate in the case Fo n g'l # O and still not sufficiently investigated in the literature to make precise statements on {ll, Fo, F~}; in any case, to the extent that such dense set of initial data exists, our results apply in principle also to the case Fort F1 = O. 1.5. An Interpolation Result Corollary 1.5 (to Theorems 1.2 and 1.3). Let either Fo # Q or else Fo = 0. Let the triplet {1), Fo, F1} satisfy assumptions (H.1) = (1.2) and (H.2) = (1.3). Let - A be the Laplacian with homogeneous boundary conditions as in (2.1) below. Then there exists T ' > 0 , such that if T > T " then, for any (yO,y l ) ~ ~(A°-O)/2)x[H~o([~)] ' (see (2.5b) and (2.5c)), 0 < 0 < 1 , there is a suitable vc[H°(O, T;L2(F1))] ', such that the corresponding solution of problem (1.1) satisfies y ( T ) = y t ( T ) = 0 . For T " we can take the same T'm stated in Theorems 1.2 and 1.3.
2.
2.1.
A Priori Inequalities
Preliminaries. Case Fo # Q Versus Case Fo = O
Let A: L2(II) D ~ ( A ) -->L2(fl) be the operator defined by Af=-Af,
~ ( A ) = { f e H 2 ( l l ) : f l r o = O - -Of ~ r=O}.
(2.1)
Then A is a nonnegative self-adjoint operator with compact resolvent R(A, A). We distinguish two cases.
252
I. Lasiecka and R. Triggiani
Case (a). Let Fo ~ Q. Then A is actually positive, since the problem Af =O,
f ~ ~(A),
i.e., (2.2)
Af=0
inn,
f l F0 =0,~
i~l=0
implies f = 0
by the Green theorem applied to 0 = (Af, f)L2~n). Thus, in this case A-' ~ ~(L2(II)).
(2.3)
Consider the space ~(A1/2), domain of the positive operator A 1/2, topologized as usual by 2
Ilzlt~A,J2)-- Ila1/Zzlt22<~)=(az, z)L~n),
z ~ ~(A1/2).
(2.4)
We have
"~(A 1/2) = H~o(f~) with equivalent norms, 2
IIz It2(A1/2)
= Ilal/2z I12=(,~= fa IVzl2 dO.
(2.5a)
In fact, for z6 ~(A), we have by (2.1) and the Green theorem
(Az, z)L2¢~)= In Ivz]2 dO.
(2.6)
The above identity can be extended to all z ~ H~-o([l) defined before, yielding H~-o(fl) c @(A1/2). On the other hand, @(A) c H~.o(ll ) and by interpolation ~ ( a ~/2) = [~(A), LZ(II)]a/2 = [H~o(II), L2(fl)],/2 = H~o(l-I ). Moreover, writing z=A-~/2A~/Zz, for z c ~(A ~/2) we obtain the generalized Poincare inequality
all
L~(a)= Cp
(2.7a)
z~(a~/~)=n~-o(t~), C~=[IA-V2[[,
to be often invoked.
Case (b). Let F o = O . (Neumann problem for (1.1).) Then A in (2.1) is not invertible on L2(f~), but is invertible (with bounded inverse) in the space L 2 ( , ) = L 2 ( f~)/ N (a )={ f ~L2 (f l ):If d l l =O),
2 The graph-norm of ~ ( A 1/2) is the same as the H~-0(l))-norm.
(2.8)
Exact Controllability of the Wave Equation with Neumann Boundary Control
253
where N(A) is the null space of A, spanned by the normalized constant function in O. We have { L2(O) = L~(I~) + )¢'(A) z= e+e,
(orthogonal sum),
(2.9)
e~ LoZ(O), e = const, e )¢'(A).
We then introduce the quantity
JR(z)=-
1
meas l)
IozdO=
1 l)(z' 1)a meas
(2.10)
so that, in the notation of (2.9), we have (~, 1)o = 0 and ~ = ~t(z); thus z ~ Lo2(f~) ¢:> {z ~ L2(f~) and ~ ( z ) = 0}.
(2.11)
The counterparts of (2.5a)-(2.7a) are now
{ {
@(A a/2) = {f~ H~(f~): ~ ( f ) = 0} with equivalent norms, (2.5b)
Ilfll2(a,/2)- IlA'/2fll~(a)= fin Ivfl 2 dO, ~ ( ( a + l ) 1/2)=Ha(O)
with equal norms,
Ilfll~'(.) = Ilfll~((a+,)"~) = ((a+ I)ff)L~(m
A= ,/2fll~=(.~2 -- C vzflvflzdO, ilfl[~(.)_< Evil da
2.2.
~l(f) = o, r
(2.5c)
J . Ifl2+lvfl 2 dO. ~t/(f) = O.
(2.7b)
Preliminary A Priori Inequalities
We shall see in Section 3 that exact controllability for problem (1.1) on the space H~o(O) x L2(O), or else on the space LZ(l)) x Hrot(O), etc., is equivalent to certain inequalities for the associated homogeneous problem ~o,, = A~o in Q, ~o(x, 0) = q~o,
(2.12a) ~o,(x, 0) = i
in 12,
(2.12b)
~o---0 in Eo,
(2.12c)
0--~-0 0v
(2.12d)
in El,
whose solution with ~o°, ~l ~ Hbo(O) x L2(O) is given by
ff~(t) = C( t)~o° + S( t)~o 1 = C(t)~o°+S(t)~+~l(~o°)+td/l(q~l)6 C([0, T]; n~o(O)), ~o,(t) = C( t)~ 1- AS( t)~o° 6 C([0, T]; L2(O)),
(2.12e) (2.12f)
where according to (2.9) we write ~00 = . ~ 0-~ d~/(~ 0),
~01 ~-,-,~~O - 1 ~- ~/[ (~01 )
(2.12g)
254
I. Lasiecka and R. Triggiani
and where C(t) is strongly continuous cosine operator generated by - A and S(t) = ~o C(r) dr, t ~ R. Note that ~t(¢ °) = j/(q~l) = 0 implies ~ ( ¢ ( t ) ) -~ 0 from (2.12e). Accordingly, we find it convenient to assemble these inequalities in the present section, for easy future reference, by showing under what conditions (typically on the geometry of the triplet {f~, Fo, F~}) they hold true. The main goal of this subsection is to prove the following three results. Theorem 2.1.
Let either Fo # O or else Fo # O. I f Fo = O, assume further that ./R(~ °) = ~t(q~ 1) =0 (cf. (2.10)). Let the triplet {1), Fo, F1} satisfy the geometrical hypotheses (H.1)= (1.2), (H.2)= (1.3), and (H.3)= (1.5), in terms of a suitable vector field h(x)¢ C2((1), as in the statement of Theorem 1.1. Then there exists Tm> O, to be specified below, such that for T > T,, the following inequality holds true for problem (2.12a-d): (2.13)
opt dV~l >- Ch,o( T - Tm)E (O), I
E(O) = [ Iv °l=+ I 11=
equal to the ~ ( A 1/2) x L2(f~)-norm of {q~o, q~l} ( cf. (2.5a)), in turn equivalent to its Hlo(f~) x L2(f~)-norm; (2.14)
Jfl
for all ~po, ~ in H~o(fl ) x L2(fl) for which the left-hand side of (2.13a) is fnite. Moreover, for T,, and Ch,p in (2.13a) we can take Tm -
2Kh
p -2G~c~
F Cp,
Kh = DhCp + Mh,
(2.15a)
p - 2GhCp sup Ihl ' r~
ch,~
Mh --~max Ihl,
(2.15b)
2Dh = m~x[div hi,
4Gh ~ max[V(div h)l.
(2.15c)
Theorem 2.1 will follow from the following. Lemma 2.2. Let {[/, Fo, F~} satisfy the geometrical assumptions (H.1) = (1.2) and (H.2) = (1.3) in terms of a suitable vector field h(x)~ C2(ffl). Then the following inequality holds true for the solution of problem (2.12a-d): ½f~ ~ h . vd21>-p f Q l v ~ , 2 d O + ½ I Q ~ V ( d i v h ) . V ~ d Q + f l o , r ,
(2.16)
I
where flo,T (boundary term at t = 0 and t = T) is given by 1 /3o.r = ~[(q~,, ~Pdiv h)a]or+ [(~pt, h" V~p)a]or.
Moreover, if Fo# 0 , or else if Fo = O but ~ ( ¢ o ) = ~ ( 1 ) (2.15c))
(2.17a) = 0, then (see (2.7a, b),
[DhC. + Mh]E(O) = KhE(O), Fo~O;
or F o = O
and
~t(~°)=~t(¢l)=O.
(2.17b)
Exact Controllability of the Wave Equation with Neumann Boundary Control
Proofof Lemma 2.2. We us a multiplier technique as in [LLT]. Step 1. With h(x) the assumed vector field, we multiply both sides
255
of (2.12a)
by h. V¢. Proceeding as in [LLT] we obtain the following identity:
La,(h.V~)d~+½ L --
~
~,h'v
dZ-½
L
IV~l:h..dX
fq HV~'V~dQ+½ fO [~2-IV¢[2]
=
div
hdQ+[(~"h'V¢)L~(m]r' (2.18)
which we write here without using boundary conditions for easy future reference to various cases. Here, H = H(x) is the matrix defined in (1.4). To make this paper self-contained, we provide a derivation of (2.18) in Appendix A.
Step 2. To estimate the second integral on the right of (2.18), we multiply both sides of (2.12a) by ~ div h and integrate by parts. Using the identity V~o. V(~ div h) = ~V(div h). V~ + IV~[ 2 div h
(2.19)
we obtain
fQ[~,~-Iv~l~]div h dO =
L
~V(div h) • Vq~ dO-
L~q~
div h d £ + [(q~,, q~ div h)Lqm] r,
(2.20)
which we again write without using boundary conditions. For future reference, we note that the above argument yielding identity (2.20) holds for any smooth vector field, not only the postulated h. Specializing to div h -- 1 (i.e., multiplying (2.12a) simply by ~) gives the identity
Ion,~-/v~l 2dQ [(¢,, ~)L2.~]0L
(2.21)
=
to be invoked below.
Step 3.
We now use the boundary conditions (2.12c, d). Thus
Hence, combining (2.18) and (2.20) and using (2.12d) and (2.22) yields
½
~ 0
h'vd~o+½ ~2h.vdL;,1-½ [V~[2h.vd£, 1
d~l
= foHV~'V~ dO+½fo~V(div h)" V~,odO+ flo,r
(2.23)
256
I. Lasiecka and R. Triggiani
with /30,7- defined by (2.17a). Using assumption (H.2)= (1.3) in (2.23) yields (2.16) as desired. It remains to prove estimate (2.17b) when Fo~Q. First we recall the standard result that
f Iv(t)F+
E(0)
E(t) =--
(2.24)
for the conservative problem (2.12). (Multiply (2.12a) by q~, and integrate by parts.) To handle the first term in (2.17a), we use the Schwarz inequality and Poincare inequality (2.7a, b) (the latter justified since Fo # Q, or else Fo = ;~ but ~/(~o) = ~/(q~l)= 0, so that ~/(q~)= 0) and we obtain 1[(~,, ~Pdiv h)a]or[-< 2DhCpE(O), Fo#O;
or
Fo=Q,
but
.ff//(~00)=J~/~(~01)=0,
(2.25a)
see (2.15c), (2.14), and (2.24). Now applying Schwarz inequality to the second term of (2.17a), we obtain the estimate (see (2.15c), (2.14), and (2.24)) I[(qb, h. V~p)a]oTI- MhE(O),
(2.25b)
which is valid in both cases FoCQ and Fo=Q. Then (2.25a, b) yields (2.17b). [] Proof of Theorem 2.1. Step 1. We return to inequality (2.16). With reference to the second integral on the right-hand side (R.H.S.) of (2.16), we have
l lnq V(divh)'V d >--2Gh IId Ivd da =-2Gh(Id,Ivd)Q,
(2.26)
recalling (2.15c). Since Fo ~ 0 , the generalized Poincare inequality (2.7a) holds and yields
-2Q(Id, Ivd).--2G~ll~ll IllVdll>--2Qcp
f, iVd=da.
(2.27)
Thus, using first (2.27) in (2.26) and then (2.26) in (2.16), we obtain ~,h. ~,d~l--- R.H.S. of (2.23) 1
(P - 2GhCp) IolVCp[2dQ - KhE (0).
(2.28)
Step 2. We now recall identity (2.21). Since Fo ~ •, or else Fo = Q but ~(¢po) = 2/(q~ 1) = 0 so that ~(~o) = 0, we obtain by Poincare inequality (2.7a, b) and (2.24) (as in (2.25a))
Fo#Z;
Fo=O,
and
d~(¢°)=~(¢1)=0.
(2.29a)
Exact Controllability of the Wave Equation with Neumann Boundary Control
257
Hence
Fo~O;
or
F0=~,
and
~/(~°)=~t(~l)=0
Fo=~
and
~(~°)=~(~)=0.
(2.29b)
and, recalling (2.24),
Fo#O;
or
(2.30)
Under assumption (H.3) = (1.5), then (2.30) inserted in (2.28) gives
~,h.
~, d : L -~ [ ( a
-2GhC~I(T-Cp)- 2 K ~ ] B ( O )
1
=(p-2GhCp)[T-(
2Kh +Cp)]E(O). p -2GhCp
(2.31)
Then (2.31) yields the desired conclusion (2.13)-(2.15) of Theorem 2.1.
[]
An important variation of Lemma 2.2--to be crucially used in the developments of Subsection 2.3--is the following lemma which is valid for both cases Fo # O and Fo = O. Its impact is that it allows us to dispense with assumption (H.3) = (1.5) by employing property (1.11) instead. On the other hand, in the case Fo = O, Lemma 2.3 does not provide the constant C~- as a linear function of T. Lemma 2.3. Let {1), Fo, F1} satisfy the geometrical assumptions (H.1) = (1.2) and (H.2) = (1.3), in terms of a suitable vectorfield h(x)s C2(1)). (a) Let Fo ~ Q. Then there is Tm> 0 (to be specified below) such that if T> Tin, then the following inequality holds truefor problem (2.12a-d):
f ~ d~l+fo~2dO>-CTE(o).
(2.32)
1
I~h.~
\p
-- e G h
'
~h,,--= max/max Ihl, 2 G---~hl, Kh = DhCp+ Mh asin (2.15a), I. F1
Tm = 2 Kh+ Cp
EJ
(2.33b) (2.33c)
P
for any 0 < e < p/ Gh. (Compare Tm in (2.33c) with Tm in (2.15a).)
258
I. L a s i e c k a a n d R. Triggiani
(b) Now let F o = Q . There is T ' > 0
(specified below) such that if T> T ' , then the following inequality holds true for problem (2.12a-d):
fd C~-
C E(0). tZ'h..... (T)
T-
e1+2
!
(2.35a)
p--eGh ] l'
IX'h..... (T)=--max{m-arxlhi'2 GhT+2Dh+p(e-Ghe) Tm-
(2.34)
eI
~11}'
2Mh
(2.35b)
(2.35c)
P
With Tm given by (2.33c) (resp. T" given by (2.35c)), for any T> Tm (resp. T ' ) we select e > 0 (resp. ea > 0 and e > O) small enough to make the expression in the square bracket in (2.33a) (resp. in (2.35a)) positive. Proof of Lemma 2.3. Step 1. We return to identity (2.23) (which was derived without making use of either assumption (H.1)= (1.2) or assumption (H.2)--(1.3)). We estimate the second integral on the right of (2.23) in a way different from (2.26)-(2.27) (the latter equation requiring Fo# 0 ) : 1 2
fd~V(div h).V~ dD~_>-2Q(J~I,Ivd). >>--a~ f l~12df~-b~ f [v~12 df~
(2.36)
for any e > 0 where
a~ -
Gh 6
,
b~ = eGh.
(2.37)
Now invoking assumption (H.2)= (1.3), we obtain, for the right-hand side of (2.23) via (2.36),
R.H.S. of (2.23)>-(p-b~) fQIV~12 dQ-a~ fQ¢2 dQ+/3o-r
(2.38)
for e sufficiently small. We note explicitly that (2.38) is valid in both cases Fo # Q and Fo = Q.
Step 2. To estimate /30,7- we recall that its second term in (2.17a) is likewise upper bounded in both cases Fo # • and Fo = ~ in (2.25b). It is at the level of estimating the first term in (2.17a) that we must distinguish between Fo ~ O and Fo = Q. The estimate for Fo # O was already obtained in (2.25a), using the Poincar6
Exact Controllabilityof the Wave Equation with Neumann BoundaryControl
259
inequality. Instead, for Fo = O, we write by Schwarz inequality
][(~t, ~ div h),]orl -
DhIt el(ll ~pt(T)II 2+ ]l
ill2)+1 (ll ¢p(T)II2+ H~pol]2)/ ) E1
<_2Dh[elE(O)+lIIq~II2c([O,T];L2(n)~]
(2.39)
in place of (2.25a). Thus, putting together (2.17a), (2.25b), and (2.39), we obtain, for Fo = O (but also for Fo # 0 ) ,
I#o,rl- [exOh-kgh]E(O)+1 Ohll~II (to,TJ;L2(->>,
(2.40)
E1
in place of (2.17b). Thus, using (2.38) and either (2.17b) or (2.40), we obtain the following estimates for e sufficiently small: for Fo @0 ,
R.H.S.of (2.23)>-(p-b~)fQ,V~pI2dQ-a~ Io~p2dQ-KhE(O);
(2.41)
for Fo = 0 , R.H.S. of (2.23)--> ( p - b . ) f o l V ¢ I 2 1
--oEl 11
dQ-a. fQ 2dQ-[e,Oh+Mh]E(O)
I] C([O,T];L2(y'Q).
(2.42)
Step3.
Let first Fo ~ 0 . We use (2.30) in (2.41). Recalling the top line of (2.28) we obtain ½f ¢2h. ~,dEl-> R.H.S. of (2.23) dE 1
>-(p-b~)(T-~)E(O)-KhE(O)-a~fo~p2dQ.
(2.43)
Instead, let Fo = 0. We return to identity (2.21) and we estimate as in (2.39)
I f o ~ - 'V~[2 dO l = [[(~'' ~ )~]~[<-elE (O)+ l 'l~p
(2.44)
in place of (2.29a). Hence
[IV~pI2dQ>-[~dQ-elE(O)-II[~II2c([o.T];L2(~)) for El Jo Jo
Fo=O,
(2.45)
and recalling (2.24)
fo[V~[2dQ>-½folV~p'2dQ+½fo~ dQ-Xe~E(O)--~e[[~P[[2c(to.Tl;t?(~a)) =(T--½el~E(O)-~l--~---II¢II2c(to, T];L~(m) \z l ze~
for
Fo=O.
(2.46)
260
I. Lasieckaand R. Triggiani
Using (2.46) into (2.42) and recalling the top line of (2.28) results in ~th" ~'d~l-> R.H.S. of (2.23)
(2.47)
1
1 2 --a~ fQ~2dQ--[-~l1 Dh+(p--b~)-~e]]]'q~[lc(to, T];L'(C~)).
(2.48)
Step4.
For Fo ~ O, inequality (2.43) easily implies (2.32)-(2.33a, b, c), via (2.37). Similarly, for Fo = O, (2.47) implies
(n~xlhl) fE 2dy.+[2a~T+2Dh+(p_b~) 1 ] ~ (fl - be)[T-( el~-2 81D~-bMh)]E(O)
(2.49)
from which (2.34)-(2.35a, b, c) follow. The proof of Lemma 2.3 is complete. []
2.3. Absorptionof Lower-OrderTerm lt~ll~(to,~l;,>(,~>>. Another A PrioriInequality We recall that with Fo # O the last step in the proof of Theorem 2.1 (from equation (2.26) to conclusion) consists in "absorbing" the interior term II ll by the energy term IIIv l II by use of Poincare inequality (2.7a) in equation (2.27): the price paid in this approach is the requirement of the additional (and undesirable) assumption (H.3) = (1.5) at the level of obtaining (2.31). In this subsection our starting point is Lemma 2.3, thereby we assume only hypotheses (H.1) = (1.2) and (H.2) = (1.3) and, moreover, we also consider the case F0 = ~. Then Lemma 2.3(a) and (b) for Fo ~ O or Fo = Q (with no assumption ~(¢o) = 0 needed), respectively, lead to an inequality like (2.34), which we rewrite here for convenience as
a priori
+ II¢ II
c->>-
(2.50)
(.for all T > s o m e 7"1>0, for either case F o ~ O or Fo=O.
T,=I2DhCpp+Mh+Cp=Tm in(2.33c) t2~=
T" in (2.35c)
if Fo~O, if Fo=O,
where the positive constant C~,r depends on T but not on ¢o, ~ . Indeed, C~,T coincides with C~ given by equation (2.35a) in the case F0 = Q; while C~,T is given by Cr/max{1, T} in the case Fo#O, where Cr is defined by (2.33a).
Exact Controllabilityof the Wave Equation with NeumannBoundaryControl
261
Moreover, T~ coincides with Tm given by (2.33c) in the case Fo = Q, and with T'm given by (2.35c) in the case Fo = 0 . This section provides another more sophisticated approach to the problem 2 2 of "absorbing" ~o 2 dQ in (2.36a) or I[~llc¢o,~j;~ <,>> in (2.50). This approach is based on compactness arguments of the type also used in exact controllability questions by Littman ILl 1] (who invokes H/Srmander) and Lions [L8] (who uses a remark of P. L. Lions). This line of argument manages to dispense entirely with assumption (H.3), by eventually relying on the uniqueness property (1.11). Lemlna 2.4.
Assume that the solution ~ in (2.12e) of problem (2.12a-d) with o , ~Pl in H ~o(fl ) x L 2(ll ) satisfies inequality (2.50) for T > T1 > O. ( This is guaranteed to hold true under assumptions (H.1) and (H.2) by virtue of Lemma 2.3 in either case F o # Q and Fo=O.) Then, for all T>max{T1, Tu}, 7"1 defined by (2.50), Tu by (1.11), there exists a constant C~,r> O, depending on T but not on the initial data, such that 2 ~ t f ~2+~2dE~, II~llc~tO,~l;~<,~>>-c~,~ d~
T>max{T1, T~}.
(2.51)
t
For easy reference, we state as a separate result the following immediate corollary. Theorem 2.5. In both case Fo# O and Fo=O, assume hypotheses (H.1)= (1.2) and (H.2)= (1.3) on the vector field h (so that inequality (2.50) holds true), so that inequality (2.51) holds true. Then, for all T>max{T1, Tu}, T1 defined by (2.50), Tu by (1.11), the solution q~ given by (2.12e) of problem (2.12a-d) satisfies (inequality (2.51) and hence) the inequality
Is 2
2
c1,~
(2.52)
~Pt +~p dXl--> C~,T+ 1 E(0),
where the constants CI.T and C ~.T are the same as in (2.50) and (2.51), respectively. Proof of Lemma 2.4.
The proof is by contradiction.
Step 1. Suppose there exists a sequence {¢n (t)} of solutions to problem (2.12a-d), i.e., r
~" =A~n
in Q,
(2.53a)
~n[t=O ~
0 1 ~n~Hro(l-l),
~ nt l t = o = ~ L 2 ( l l )
¢,-0
in Eo,
(2.53c)
0~°"---0 in E~, Ov
(2.53d)
in ll,
(2.53b)
over [0, T] explicitly given by ~on(t) = C ( t ) ~ ° + S(t)~o~6 C([0, T]; Hlo(12))
(2.53e)
262
I. Lasiecka and R. Triggiani
such that, with d / dt = ',
I
ll ~, IIc~to.T1~L2C", = 1, 2
t 2
f q~, + (~o,) d]£1-->0 kaz~
(2.54a) as
n-->oo.
(2.54b)
By assumption (Lemma 2.3), the solutions ~o,(t) satisfy inequality (2.50) and by (2.54) we have
E.(0) = II~ (~.)1 2 +lv~.l02 df~-const.,
uniformly in n.
(2.55)
We can thus extract a subsequence, still subindexed by n, such that (V q~° converges to some function in [LZ(fl)] aimn weakly, and hence) there are constants c, for which
[
~o° + c, --) some function (po in HI(fD weakly,
~ol,--) some function ~oI in
L2(~'~)weakly.
(2.56a) (2.56b)
If Fo # O, the condition ~o,+c,~° H~o(12) implies, by (2.53b), o + O=[~°+c.]lro=~.lro c, =
c..
(2.56c)
If Fo = ;~, we may assume without loss of generality that q~o e L2(fD, i.e., ~/(q~o) _ 0, see (2.8) and (2.11), i.e., that M(q~°+c,) = c,. Step 2. (Solutions to problem (2.12a-d) with initial data as in (2.56).) If Fo = Q, it is convenient to split quantities in two orthogonal components in L2(12) and in ?((A), as in (2.9). Thus set 1
1
~1 = ~01--~ cl '
q,o = q,o + c o,
~o~, ~o1, ~ ° c L~(fl),
Cln, C1, c° e )C'(A),
(2.57a) (2.57b) (2.57c)
since L~(12) is invariant under C ( t ) and S(t), we have C(t)(~p ° + c.) = C(t)~o ° + c.,
(2.58a)
s( t)~ 1. = s( t)~'. + 4 t,
(2.58b)
C(t)ff °= C( t)~° + c °,
(2.58c)
s ( t )~, ' = S ( t ) ~ 1 + cl t.
(2.58d)
Thus, the solutions q3,(t) due to initial data [~o°+ c,, ~ol,]
Exact Controllability of the Wave Equation with Neumann Boundary Control
263
and ~0(t) due to initial data [~O°, ~o1] of problem (2.12a-d) are given by
¢pn(t) = C(t)(~O°n + Cn)+ S(t)~Oln ~"~,(t)
for F o ~ Q (see (2.53e), (2.56c)),
(2.59)
=[~.(t)+c.=C(t)~°+s(t)--~.+co+~t for to=O, ~'(t) = ~o'(t)= - A S ( t ) ~ ° + C(t)~l,,
{
see (2.53e),
(2.60)
~(t) = C(t)O°+S(t)~ 1,
(2.61)
~'( t) = - A S ( t)~° + C ( t ) ~ 1.
(2.62)
Step 3. It follows that
{
~,(t)->~b(t)
in L°~(0, T; H~o(fl)) weak star, ff'(t) = q~'(t)--> ¢'(t) in Lr~(0, T; L2(I"I)) weak star.
(2.63) (2.64)
In fact, with reference to
{
f f , ( t ) - ~ ( t ) = C(t)(~°-O°)+S(t)(q~l-q~l)+(cl-cl)t+(c,-co),
(2.65)
~" tn ( t ) - ~ (tt ) = - A"
(2.66)
1/2
S(t)A
1/2
( ~ 0n - ~- ' ~ ) + C ( t ) ( ~ n1 - ~~ ) + ( c--~~ - c -'~ ),
i n t h e case F_~o=~3 (in the case Fo# Q, delete the superscript "bar" on ~b°, ~o1,, q~l, and set c~, = C1 = c, = Co---0, see (2.56c)), if now
gl c Ll(o, r; [~(A1/~)]'), gEe LI(0, T; LE(fl)), then
f f (A1/2[C(t)(~ ° ~ ) + S ( t ) ( ~
l
~01)],A-1/2gl(t))m:
+ (_A1/2S(t)A1/2(o _ ~o)+ C(t)(~l _ 1), g2(t))L:~a) dt f f (A1/2(~ ° - ~ ) , C(t)A1/2gl(t)-A1/2S(t)g2(t))g~0
(2.67)
by the Lebesgue dominated theorem and (2.56), since ]]C(t)[[, I]A~/Es(t)]] < const., t in [0, T]. Then (2.67) yields (2.63)-(2.64).
Step 4. It follows from (2.63)-(2.64) and trace theory that by compactness there is a suitable subsequence such that ~,]~ = [~o, + e,]~,-> ~bt~, in, say, LE('E1).
(2.68)
If Fo # ~ , we have seen in (2.56c) that c,-= 0. If Fo = ~ , then condition q~,l~, -->0 in L2(~1) from (2.54b), combined with (2.68), gives c,-> some constant c. Thus,
264
I. Lasieeka and R. Triggiani
we have proved there is a subsequence such that ~q~o~ some q~0 in Hl(ll) weakly] ~from (2.56), ~o~,~q~~ in L2(~) weakly J
(2.69b)
~0,(t)~ some function q~(t) in L~(0, T; H~.o(fl)) weak star] from
(2.69c)
~o'(t)~ q~'(t) ,¢,lzq~q~lx~----0
in L°~(0, T; L2(I))) weak star in L2(E1),
(2.69a)
l(2"63), (2.69d)
from (2.54b), (2.69c), and (2.68).
(2.69e)
Step 5. By (2.69c), {q~,(t)} is uniformly bounded in L~(0, T; H~-o(l))) and by compactness there is a subseqence ¢, strongly convergent in L~°(0, T; L2(I))) to ~o. Thus
II tl
II II
1
(2.70)
by (2.54a). But, by (2.53e), the limit q~ satisfies problem (2.12a-d) and, moreover, by (2.69e) we have ~o[zq-=0. Thus, the limit ~ satisfies the problem
{
~"=A~ ~olx=0
0~o = 0 Ov Ix~
in Q, in E,
for T > m a x { T , , T,},
(2.71)
in Y.1.
Then, Property (1.11) applies and we conclude that ~ - - 0 in Q, a contradiction with (2.70). Lemma 2.4 is proved. [] We next present an important improvement of Theorem 2.5 in the case Fo = ~3, when E (0) is not equivalent to II{ o°
, }II.,(.)×L=<.) =
if M(q~°) # 0.
Let Fo = •. Assume, further, hypotheses (H.1) and (H.2) on the vector field h, so that conclusion (2.52) of Theorem 2.4 holds true. Then, for all T>max{T1, T,}, 7"1 defined by (2.50), T, by (1.11), there is a positive constant kr, depending on T but not on q~o, ~ol, such that the solution q~of problem (2.12a-d) satisfies
Theorem 2.6.
fx 2+u,2 dX>_kTll{~oo, q~h,,2 111-'(a)×L m).
(2.72)
(Note: we are not assuming A/(q~°) =0.) Proof For T > T,, the uniqueness property (1.11) implies that { fx~o2+ ~2 d~} '/2.
(2.73)
is a norm. Now let T > max{T~, T,}. If {~o,(t)} is a Cauchy sequence for this norm with ~o,(0)= ~o° and q~'(0)= q~,, then the conclusion (2.52) of Theorem
Exact Controllabilityof the Wave Equationwith Neumann BoundaryControl
265
2.5~which holds true under present assumptions--implies that there is a sequence {c,} such that ~,o + c, converges in Hl(f~),
(2.74a)
~
(2.74b)
converges in L2(I~),
and by trace theory 0, + c,
converges in L2(F).
(2.74c)
On the other hand, since both {~,(t)} and {~'(t)} converge in L2(£), i.e., {~,(t)} converges in H~(0, T; L2(F)), then the Sobolev imbedding theorem implies that ~o,(0) = ~o°
converges in L2(F).
(2.75)
Comparing (2.75) with (2.74c) yields that the numerical sequence {c,} converges. Thus, {¢p,(t)} being a Cauchy sequence for the norm (2.73) implies that {~o°} is convergent in HI(O) and {~o~}is convergent in L2(I'I). Thus, inequality (2.72) is proved. [] 2.4.
Absorption of Lower-Order Term ~ , Theorem 2.1 Revisited
2 d£1. A Priori Inequality of
In this subsection our starting point is Theorem 2.5 for Fo # O or Theorem 2.6 for Fo = O. Thus, we assume hypotheses (H.1) and (H.2) throughout. As a result, we obtain that, for all T>max{T1, T,}, the solution of ~0 of problem (2.12a-d) satisfies inequality (2.52) for Fo # O or inequality (2.72) for Fo = O; i.e. (we rewrite these in a combined form), f~ ,
~02+ ~2 d~"l
> rE(O) C2T ~ 0 1 2I 2 ' [JJ{~o,~o}ll.(a)×t(~)
E(0) equivalent to [[{~o°, ~'}[I 2~o(m×L2(a), T>max{Tl, T,},
for Fo~O, for Fo=O,
(2.76)
F0 # O,
T1 defined in (2.50), Tu defined by (1.11),
where C2,r is a positive constant depending on T but not on the initial data. Our next step is then to employ an argument patterned after Lemma 2.4 in order to "absorb" the term ~, ~o2 d£1 by the term ~, q~2d£~. Lemma 2.7. Assume ~o°~ Hlo(fl) and ~01EL2([-~) and, if Fo=O, assume further Jtt ( ~o°) = O. Consider the corresponding solution ~oof problem (2.12a-d) and assume that q~ satisfies inequality (2.76). (By virtue of Theorems 2.5 and 2.6, this is guaranteed to hold true under assumptions (H.1) and (H.2).) Then, for all T > max{T1, Tu}, T1 defined by (2.50), Tu by (1.11), there is a positive constant C~.r depending on T but not on the initial data such that f ~o2 d~,l <-~C~,T ~ q~2tdY.1. t
d'~l
(2.77)
266
I. Lasiecka and R. Triggiani
For easy future reference we state as a separate result the following immediate corollary. Theorem 2.8. In both cases Fo# Q and Fo = 0 , let ~o°, q~l ~ H~o(l-l) x L2(f~) and, /f Fo = Q, let further ~ ( o) = O. Assume hypotheses (H.1) and (H.2) on the vector field h. Then, for all T > max{T~, T,}, the solution ~o given by (2.12e) of problem (2.12a-d) satisfies (inequality (2.77) and hence) the inequality for Fo#0,
I fJx, q ~ d X , - > I S C ~r 2 =.~ t[E(0~ I1{~ ,~ , }tl..,)×L ~.) for Fo=~5; d//(~p°)--0, (2.78) E(O) equivalent to t1{~°, d}ll~,to.~)×m~),
Fo:# Q,
T > max{T1, T,}, where the constants C2,7rand Ct2,T are the same as in (2.76) and (2.77), respectively. Proof of Lemma 2.7.
Same ideas as in Lemma 2.4.
Step 1. Suppose, by Contradiction, that there exists a sequence {q~,(t)} of solutions as in (2.53a-d) such that
11~o. IIL~,) = 1, I~ (q/)2 d E l ~ 0
(2.79a) as
n~a3.
(2.79b)
By assumption, the solutions ~,(t) satisfy inequality (2.76) and thus the pairs {~o°, ~o~} are uniformly bounded in Hlo(12) × L2(I)), as in Step 1 of Lemma 2.3, in both cases Fo # Q and Fo = Q. It then follows, as in Steps 2-4 of Lemma 2.3, that for a suitable subsequence we have
{
q~,(t)~some ~(t) in L~(0, T; H~o(~)) weak star, qY(t)~ ~o'(t) in L~(0, T; L2(f~)) weak star.
(2.80a) (2.80b)
By (2.80) and by compactness, then for a suitable subsequence we have ~,lx, -->9Ix, in L2(•1) (strongly)
(2.80c)
and by (2.79a) we deduce
II~ IIL2~,) = 1.
(2.80d)
On the other hand, (2.79b) implies q~'lx, =0.
(2.80e)
Exact Controllabilityof the Wave Equation with Neumann Boundary Control But, by (2.53e), the limit function ~ satisfies the problem
Differentiating in time (2.81) and using (2.80e), we obtain that ¢ ' = ~0t solves
Then, property (1.11) applies and we conclude that ¢'---0 in Q. Thus, ~o---const. in Q. If Fo # ~ , then (2.81b) yields ~o-= 0 in Q. If Fo = @, the further assumption d~(¢°) = 0 yields likewise ~o--0 in Q. Thus in any case, the conclusion ~o-= 0 in Q contradicts (2.80d). Lemma 2.7 is proved. []
Throughout this subsection we take F0 = O (but we do not assume M(q~°) = 0) and we specialize the vector field h(x) to a radial field x - x °, for some fixed Xoe R'. Thus, recalling Remark 1.1 and (2.15c)
In the main statement below we need the tangentialgradient VMb of a function ¢ ~ Cl(fi) on F (or part thereof; in our case the set F . ( x °) defined in (1.14)). At each point of F (sufficiently smooth), consider the unit outward normal 1, and a, say, orthogonal system of unit vectors Z l , . . . , zn-1 on the corresponding tangent plane.
This, if O@/Or,=O on F we set
and we then define in this case
268
I. Lasiecka and R. Triggiani
Our main result in this subsection is the following inequality which requires no geometrical conditions on fL Theorem 2.9. Let Fo = • and recall the sets F(x °) and F.(x °) of F defined by (1.14). There exists Tm >0 (given explicitly below) such that for all T> Tin, the solution ~ given by (2.12e) of problem (2.12a-d) with Fo = Q satisfies the following inequality: F(x )
0
1
2
J0
./F,(x )
./0 dF
c~ll{~, ~ }II.'~.)×L2~.) (2.87) for all {~o°, ~pl}c H i ( o ) x L2(f~) for which the left-hand side is finite, where Cr is a positive constant depending on T but not on 9 °, ~ l. For Tm we can take (see (2.83)) Tm = max{ T,, 2R(x°)}, (2.88) where T~ is the time defined by the uniqueness property (1.11) with Fo=O, see Remark 1.3. ---
Proof of Theorem 2.9. Step 1. (Variation of Lemma 2.3.) Lemma 2.10. For Fo=Q we have, for any e > 0 and m ( x ) = x - x °,
f ~ Ir(xO)q~m(x)'v(x)dF(x°)dt+ fTo lr,(xo)lVq~12[m(x)[dF*(x°)dt +(l+n) E
II~ II2ccto,~l;L2~.))~ ( T - L ) E (0),
T~=e(l+n)+2R(x°),
~°, ~ ~ nl(fl) x L2(f~).
(2.89) (2.90)
Proof of Lemma 2.10. (Specialization and variation of proof of Lemma 2.3.) Under specialization (2.83), the fundamental equality (2.23) for problem (2.12a-d) becomes then ½I ~2m. vdY.-l l IV~12m.~,d~= IolV~12 dQ+flo, r,
(2.91)
where flo,7-is given by (2.17a) with h(x) = re(x). We obtain from (2.40) and (2.83)
BO.T> 2 T];L2(f~))" - - - - [ 2 e+R(x °) ] E(O)- 2-~E1[~p11C([0' Thus, (2.91), (2.92) give
(2.92)
Exact Controllability of the Wave Equation with Neumann Boundary Control
269
As to identity (2.21), by proceeding as in (2.39), we obtain, for Fo = 0 , ~p2_ IV~ 12
dQ
= I[(~,,
e II~ II
(2.94)
and hence
I91V~12dQ>_fo~2dQ-[eE(O)+lll~ll2(to.r,;L2(a))]
(2.95,
in place of (2.29b). Thus, by (2.95) and (2.24), proceeding as in (2.30) we obtain, for Fo = Q,
__
(T-e,E(0) 2
-
-
1
11 ll2
C([O,T];L2(•))
(2.96a)
in place of (2.30). Inserting (2.96) in (2.83) yields
f: ~2tmvdE- I. 'V~pl2mt.'d~ +( m~+l)[[~ll2([o,T];L:(a)) >--( T- T~)E(O)
(2.96b)
with T~ as in (2.90). We now split Sr = Sr(x°)+~r,(x°), recall definition (1.14), drop negative terms, and then (2.96) yields (2.89). []
Step 2.
(Absorption of
Lemma 2.11.
II~ll~to,~;~(.,
as in Lemma 2.4.)
Let F o = ~ and let T> Tin,see (2.88). Then
H~H2c~to,r~;L~(~))<-C'r{Iro lr~ O)~p~dr(x°) dt + for fr.( o)lV~[2dF*(x°) dt+ Ior lrq~2dF dt} • (2.97)
Proofof Lemma2.11.
By contradiction, as in the proof of Lemma 2.4, let { ~ ( t ) } be a sequence of solutions (2.35) such that [1~ [[cqo, r];L~(n)) -= l,
(2.98a)
(2.98b) Each ~p,(t) satisfies (2.89). By taking e sufficiently small, we then have from (2.89) that if T > T,, then E, (0) _
270
I. Lasieckaand R. Triggiani
Lemma 2.4 then yield that, in the notation of Lemma 2.4, the limit function satisfies 1 ----II~ IIL~
(2.99)
as well as
{
~ " = A~o i n Q ,
°~°=0 ov
in£.
But the last integral term in (2.98b) gives likewise ~o=0
in£.
T=, the uniqueness property (1.11) then yields ~ -
For T > of (2.99).
Step 3.
0 in Q, a contradiction []
Putting together Lemmas 2.10 and 2.11, we have immediately:
Lemma 2.12.
For Fo = •,
e > 0,
and T> T~,
[(l+n)c'r+r(x°)](ffIr~xO)'pdF(x°) ~ dt+I? fr,(~o)'V'PdF*(x°) l2 +(1 +n ) e C~- I ~ 2
d£>--(T -
T~)E(0),
(2.100)
where r(x °) = maxr Ix - x°l. Step 4. Since a~/av = 0, then V~ can be replaced by V~q~, see (2.85)-(2.86).
To prove (2.87) it remains to show that E(0) in (2.100) can be replaced by }lt,,'
(ff fr( O)~2dF(x°) dt+ ff lr,< O)'V~'2dF*(x°) dt+ If fr~2 dF dt)l/2 (2.101)
norm for
is a instead of then { o } inequality .
T > T=. Thus, the proof of Theorem 2.6 using this time (2.100) (2.52) shows that if {~,(t)} is a Cauchy sequence for the norm (2.101), and (~,} are convergent in H~(fl) and L2(fl), respectively, and (2.87) follows. []
Exact Controllability on H~o(f~) x L2(f~) with Controls v E L2(0, T; LE(FI)) - L2(£I). Equivalence to A-Priori Inequalities. Ontoness Approach of the Solution Operator
For the sake of clarity of the exposition, we treat the two cases Fo ~ ~ and Fo = Q separately. Even though the conceptual approach is the same in both cases, there are a number of technical differences that arise between them.
Exact Controllability of the Wave Equation with Neumann Boundary Control
271
3.1. The Case Fo ~ O
The goal of this subsection is to prove the following result. Theorem 3.1.
x L2(f~) over the(a) Problem (1.1) is exactly controllable on the space Hro(fl) 1 time interval [0, T], 0 < T < co, by means of L2(E1)-controls v if and only if the following inequality holds: 9 ~,as, >_C~E(O)= C~11{9°, 9
}11~(A'J2)×L2(~>,
I
E(O) equivalent to
11{9°, 91}ll~o(,)×L2(m
(el (2.5a)),
(3.1)
for all {9 °, 91} ~ H~o(~) × L2(f~) for which the left-hand side of the above inequality is finite, where Cr is a positive constant depending on T, but not on 9 °, 9 ~, and where 9 solves the homogeneous backward problem
9 , = A9
on Q,
9 ( ' , T) : 9 °,
9t(', T)
in f/,
= 91
9 ----0 in 3~o, 09=0
(3.2)
inY.~,
,c3v
which is the time-reversed version of the forward problem (2.12a-d). (b) Inequality (3.1) is, in turn, equivalent to the following inequality for the same problem (3.2): forfr92d~,l
Cr']{9°,
'
2
1
Hrlo(O)-norm equivalent to the [~(Al/2)]'-norm given by [[zllt~(a'/2)l' = [[A_,/2zIIL2(,~
(3.3)
for all {9 °, 9 ~}~ L2(f~) x Hrl(f~) for which the left-hand side ,of the above inequality is finite. Proof of Theorem 3.1. Step O. We introduce the operator N: continuous Le(F0 ~ H3/2(f~) by setting (recall (2.2))
w = ]Qg
Aw = 0
in f~,
W[ro=O
on Fo,
I
[Or' Ir,
(3.4)
onr,
Let ]V* denote the adjoint operator of ]~r: (_K/v, u)L~(~) = (v, ]Q*u)L~(r,). We need the following lemma, in the style of IT3].
272
I. Lasieckaand R. Triggiani
Lemma 3.2.
For f e ~(A), we have on F,, on F0.
~/,Af={f[r ,
(3.5)
Proof of Lemma 3.2. For g s L2(F) we complete by the Green second theorem (subscripts denote L2-norms) -(]V*Af, g)r = -(Af,/Vg)n = (Af 2~rg)a
:(fA('Ng))n+(~
'lQg r - \
---~v ]r =-(f'g)r''
(3.6)
since flro = of/avir~ = 0 by (2.1); (/~g)[ro = 0, ONg/Ov[r, = g, and tL(~g) = 0 in f/ by (3.4). Then (3.6) yields (3.5). []
Step 1. As in [T2], the solution to problem (1.1) can be written abstractly by means of the following "variation of constants" formula. Let yO= yl = 0 in (1.1) and denote the corresponding solutions by y( t; yO = yl = O) =-y( t). Then s( r - t)lVv( t) Y( T) I = SFTV= y,( r)
(3.7)
C( T - t)JQv( t) at
where C(. ) is the strongly continuous cosine operator generated by - A in (2.2) and S(t)=~to C(r) dr. The operator LfT in (3.7) with domain ~(~T) = {V E L2(~I): [ y ( r ) , y,(T)] ~ H~o(ll) x L2(f~)}
(3.8)
is an unbounded, densely defined closed operator. See regularity theory in [LT3].
Step 2. By time reversibility, exact controllability of problem (1.1) at time T on the space H~o(f~) x L2(fl) by means of controls v ~ L2(E1) is equivalent to ontoness of ~T" ~T:
L2(~,) ~ ~(37T)
onto
, H~0(fD X L2(y~).
(3.9)
Let ~ * denote the adjoint operator of ~ r : (LPrg, Z )~A")×L2(~) = ( g, Y~* Z ) L2(XO.
Then, the ontoness property (3.9) for 3~r is equivalent to the property that 3~* has a continuous inverse [TL1]; i.e., Lt*T z°]l Z1
>--C'TII{Zo, Z,}[I~(a'/b×L2(n)
L2(Xl)
for some
C~>O
and all
{Zo,Z i } ~ ( ~ * ) .
(3.10)
Property (3.10) (and hence property (3.9)) is equivalent to inequality (3.1) or inequality (3.3) of Theorem 3.1.
Lemma 3.3.
Exact Controllability of the Wave Equation with Neumann Boundary Control
Proof of Lemma 3.3. By (2.4), if V ~ ( ~ T ) from (3.7) by proceeding as in IT1]:
273
and {Zo, Zl}C~(Za*) we compute
(~TV, l:~)~(A~/:)×L2(m=(alorS(r-t)31v(t)dt'Az°)L~(a) -t-(Afro C( T-- t)]Qv(!t) dt, Zl) L2(a) = (v, _KI*AS(T- t)Azo+ ~I*AC(T- t)Zt)L2(XO. (3.11)
Thus
~P* z° I = 31*A[S(T- t)Azo+ C ( T - t)z1] Zl I = 3I*A[C(t- T ) z l - S ( t - T)Azo].
(3.12)
The solution ~o(t; q~o, q~) of problem (3.2) is ~o(t; q o, q~m)=
C( t - T)q~° + S( t - T)q~~
(3.13)
so that, invoking Lemma 3.2, we have q~(t; q o, ¢pl)lr~ =
]V*A[C(t- T)~o°+S(t-
T)~].
(3.14)
Comparing (3.23) with (3.14) yields
(~'* z°) (t)=q~(t;q~°=za'~l=-az°)lri'zl
(3.15a)
-Azo.
(3.15b)
~ 0 = Zl,
91=
Thus inequality (3.10) becomes precisely inequality (3.3) since
II(zo =
~DO}[[~(A1/2)xL2(f~)=II{~ 0, ~I}HL2(D,)x[~.~(AI/2)]'. Instead, if we differentiate (3.13) in t, we obtain q~,(t; q o, 1 ) = C(t- T)~I-AS(t - T)~ ° -A-lop 1, -71=
(3.16)
(3.17)
and by Lemma 2.2 ~,(t; ~po, 1)1rl =
]V*A[C(t-
T)~p1-
S(t-
T)A~°],
(3.18)
now comparing (3.12) with (3.18) yields m
•
Zo, ¢a = zl)[rL,
(3.19)
whereby inequality (3.10) now becomes inequality (3.1) with CT = C~. Theorem 3.1 is proved. []
3.2. The Case
Fo =
We now briefly indicate the modifications that need to be made on the arguments of Subsection 3.1 in order to treat the case F o = O . The main result now is:
274
I. Lasiecka and R. Triggiani
T h e o r e m 3.4.
(a) Problem (1.1) is exactly conti'ollable on the space Ha( fl ) × L2({)) over the time interval [0, T], 0 < T < oo, by means of L2(£) = L2(0, T; L2(F))-controls v if and only if the following inequality holds true for the solution ~ of problem (3.2) with Fo = •:
"fO T- --fFI~t)t-- (t-- T)~(q~°)12 d E >_
, 2 Cr I1{~°, ~ 1}IIHI(.)×L,<.>
Hl( f~)-norm equal to ~( ( A + I) l/2)-norm, see (2.5c),
(3.20)
for all q~o, ¢1 ~ nl([~) × Z2(~-~) for which the left-hand side of the above inequality is finite, where CT is a positive constant depending on T, but not on ~po, q~l. (b) Inequality (3.20) is, in turn, equivalent to the following inequality for the same problem (3.2):
all 0 , q~l ~ Z2(~) × [Hl(~c~)], for which the left-hand side is finite. Proof of Theorem 3.4. (Modification of proof of Theorem 3.1.) Step O. We introduce the operator N1 (translation by 1 = 1 of corresponding elliptic problem): continuous LZ(F) --> H3/2([~) defined by
~(A-1)w=0
inf,,
w = N~g <::>,[Ow t~-v = g o n F .
(3.22)
Let N* denote the adjoint (NlV, u)L,(m=(v, Nl*U)/?¢). The counterpart of Lemma 3.2 is now (see IT2]): Lemma 3.5.
For f ¢ @(A),
N* ( A + I ) f = f i r . Proof
(3.23)
As in Lemma 3.2, let g ~ La(F) and compute by the Green second theorem
- (N*(A+
I ) f g)r = - ( ( A + I ) f N,g)a
= ( ( A - 1)f, N , g ) a
av = -(f, g)r.
lr []
Step 1. In place of (3.7) we now have y(T) l
( A + I ) Ior S ( T - t ) N l V ( t ) dt , yt( T) I = ~TV = ( ( A + I ) J, C ( T - t ) N l V ( t ) dt @(~T) = {V e L2(£): [y(T), y,(T)] e H'(f~) x L2(f~)}.
(3.24) (3.25)
Exact Controllability of the Wave Equation with Neumann Boundary Control
275
Step 2. Exact controllability of problem (1.1) at time T on the space H~(I~) x L2([I) by means of L2(E) = L2(0, T; L2(F))-controls v is equivalent to ontoness of ~ r : ~?T:
L2(•) = ~ ( ~ T )
onto
) Ha(f~) x
L2(~-~),
(3.26)
in turn equivalent to
,~T ZO I1 ~CtT]'{ZO, ZI)['H'(flO×L2(I~z), Z1 L2(E) C~->0,
all {Zo, zl}~ ~(&a*T),
(3.27)
for the adjoint operator (LaTg,Z ) H t ( a ) × r 2 ( a Lemma 3.3 is now:
) = (g,
~P*TZ)L2(:Z).The counterpart of
Lemma 3.6.
Property (3.27) (and hence property (3.26)) is equivalent to inequality (3.20) or inequality (3.21) of Theorem 3.4.
Proof of Lemma 3.6. Starting from (3.24) and proceeding as in Lemma 3.3, using this time HI(I~)= ~((A + I)1/2) (same norm, see (2.5c)), we find (counterpart of (3.12)) ~,
Zo = N*I(A+ I ) [ C ( t - T)g 1 - S ( t - T ) ( A + I)zo] z1
= [ C ( t - T)z~ - S ( t - T ) ( A + I)zo]r
(3.28)
by Lemma 3.5. Writing according to (2.9) Zo= ~o+/Co,
M(~o) = 0,
At(Zo)=/~o = const, c X(A),
(3.29)
and using A/~o= 0, S(t)k.o=-tko, we have (~*l::)(t)=N*l(A+I)[C(t-T)zl-S(t-T)(A+I)go-(t-T)ko] : [C(t - T)z I -- S(t - T ) ( A + I)eo]r - (t - T)/~o.
(3.30)
On the other hand, the solution ~ of problem (3.2) is i ( t ; o , ~p~)= C ( t - T)~p°+ S ( t - T)~ 1 = C(t - T ) ~ + ~ ( o ) + S(t - T)~ 1,
o =~-6+~,
M(~--6)= 0,
(3.31)
d / ( o ) = ~ = const. ~ N(A).
Comparing (3.30) with (3.31) yields
q~o= z~,
~ = - ( A + I)zo.
(3.33)
Moreover, by (2.5c), H~(~)= N((A+ I)1/2) (same norms) and thus
II{zo,
It{-(A + 1)-1 1, 0}[IHI(D.)xL2(D.) = II{ °,
(3.34)
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I. Lasiecka and R. Triggiani
Thus, using (3.32) and (3.34) we see that inequality (3.27) becomes precisely inequality (3.21) of Theorem 3.4. Differentiating (3.31) in t ~,(t; q~o, ~01) = C ( t - T)~ ~- S ( t -
(3.35)
T ) A ~ °.
Comparing (3.30) with (3.35) gives
A~o°= (A + I)~o,
(3.37)
~o~= z~ .'
By (3.29) and (3.37) Zo= ( A + I)-~Aq~° +/~o;
Claim.
~o = (,4 + l)-lAq~°.
(3.38)
We have
Ilzoll-'<.> >- cll~ °+ ~o11~'<.>. Proof of Claim.
(3.39)
In fact
Ilzoll ~,'<.~ = (~o÷ ~o, Co+ ~o),~'(.> = Ileoll~,'<.~ + II~oll =L2(I~)
(3.40)
since, via (3.38) and (2.5c), Hl(f~)= ~ ( ( A + I ) ~ / 2 ) , (~o,/~o)o'(a) = (A~P°,/~o)L2(,) = 0
(3.41)
(orthogonality of Lo2(f~) and W(A)). Then (3.40) becomes, via (3.38),
Ilzoll~,.<.> =
II(A+I)-'A~°II~(~A+,~.,~+ II~o11~(.~
= II(m + I)-'/2a~p° II~(-)+ II~o II~(.) >- e IIml/=~P°]l 2~(.)+ II/~oII~=(-) (taking 0 < c < 1, without loss of generality since
II~°11-< c llA1/%°ll)
-- 2 2 - c{llA 1 / 2 '~ 0 IlL2 2.,)+ IlkollL (,,>}-> c{11~°11~,'<,,)+ Ilkoll~,'<,~>} = cll~ °+ &ll ~,'(.),
(3.42)
where, in the last step, we have used (¢o,/~o),'(a) = (A¢ °,/~o) = 0. Thus, the claim is proved. [] By virtue of (3.39) and (3.37) we obtain, for Zo= ~o+/~o,
q' ' }11,, , <,~×,2~,~. (3.43) Hence, if we set ~o= ~po+/~o, i.e., ~ ( ~ o ) = cO=/~o= J//(Zo) by (3.29) and (3.31), then equations (3.43) and (3.36) become, respectively, II{zo, z,}ll,.,,~.~×L2~.~
II{zo,
CIl{q,°+~o,
Z1}IIHI(•)×L2(D.) ~ CIl{~ °, ~'}IIH'(.~×L2(.~,
= [~o,(t; ~o, q~l)]lr - (t - r)~(~o°).
(3.44)
(3.45)
Thus, by virtue of (3.44)-(3.45), we see that inequality (3.27) becomes precisely inequality (3.20) with CT = C~. Theorem 3.4 is proved. []
Exact Controllabilityof the Wave Equation with Neumann BoundaryControl
3.3.
277
Completion of the Proof of Theorems 1.1 and 1.2
Let Fo # Q. Then Theorem 1.1 follows by simply combining Theorems 3.1 and 2.1. Now let Fo = O and ~ ( y O ) = 0, ~ ( y ~ ) = 0. In this case, the dual variables Zo and z, in Subsection 3.2 satisfy likewise Jg(Zo)= 0, At(Zl)=0. Then, ~(¢o)__ Jg (Zo) = 0, see below (3.43), and A/t(tp 1) = J/t(Zl) = 0, see (3.37). Hence, Theorems 3.4 and 2.1 with M(q~o)= ~ / / ( ~ ) 1 ) : 0 combined yield Theorem 1.1. [] To prove Theorem 1.2, we replace Theorem 2.1 with Theorem 2.8 in the above arguments. []
.
Exact Controllability o n L 2 ( ~ ) X [H~o(l'l)]' with Controls v e [H1(0, T; L2(F1))] '. Equivalence to A Priori Inequalities. Ontoness Approach of the Solution Operator
Again we treat the two cases Fo # ~3 and Fo = Q separately.
4.1.
The Case F o # O
Our goal is to prove the following result. 1 r Problem (1.1) is exactly controllable on the space L2(fl) x [Hro(fl)] over the time interval [0, T], 0 < T < m, by means of controls v ~ [Hi(0, T; L2(F1))] ' if and only if the following inequality holds:
Theorem 4.1.
fr~2+~2 d~,>frll{~ -o,cP J'll~(a / )xL2(C~)= CrE(O), '"'=
,=
1
E(O) equivalent to ]1{~°, ~o, }[lU~om)X 2 L2(m,
(4.1)
for all {~o, 1 } ~ Hbo(l-l) x L2(~) for which the left-hand side of the above inequality is finite, where Cr is a positive constant depending on T, but not on ~ o, ¢ a and where solves the homogeneous backward problem (3.2). Proof of Theorem 4.1. Let v E [Hi(0, T; L2(F~))] '. The exact controllability in question is equivalent to the property that the solution operator ~ r in (3.7), rewritten here as y(T)
= . . ~ r v = l A f ~ S(T-t)]Qv(t)dt (4.2)
yt(T)
IAfo C ( T _ t ) ~ l v ( t ) d t
278
I. Lasiecka and R. Triggiani
satisfies the ontoness condition
Let:
[H'(O, T; U ( r , ) ) ] ' = ~(Ler)
onto
, L2(12) x [H~o(f~)]';
or equivalently
~e*~ Zo
II
--- c~tl(zo,
" ZdllL2(.)×t~(A'/2)l'
and all
[Zo, z~]~ ~ ( ~ * ) .
Zl [HI(O,T;L2(rl))] '
for some
C~>0
(4.3)
Let A be the self-adjoint isomorphism H~(0, T; L2(F1))-~ H°(0, T; L2(FI)): (f, g)iq,(o,r;L2(r~))= (Af Ag)L2(o,r;L2(rl)),
(4.4)
ilAfll~:(o:;L~(r,) ) = ilfll%,(o:;/~(rm) ) = [ifll~=(o:;~(r,)) + df =
dt L (o,r;t. ~(r,)) ,
(4.5)
( f g)tnm(o,r,L2(r0)l, = ( A - i f A-Ig)L2(O,T,L2(r0).
(4.6)
From (4.6) and self-adjointness of A on L2(0, T; L2(F~)) we get
=
z1 LE(o,T;LZ(Ft))
A-I~ zo
.
(4.7)
Z1 ] L2(O,T;LZ(FI)) On the other hand, we compute, from (4.2), as usual
(¢'~Tvs l zz:I) L2(D.)x[~(AI/Z)],
=(a I: S(T-t)tQv(t) dt, zo)L2(n)+(f: c(r-t)lQv(t) dt, Za)L2(n) = (v, 1V*A[S(T- t)Zo+C(T- t)A-lz~])L~(o,r;L~(r~)).
(4.8)
Using that C ( . ) is even and S(. ) is odd, we obtain, from (4.7), (4.8), and (3.5),
= ~p(t;
o = A-lzl, ~
= -zo)lr,,
(4.9)
where ~ solves problem (3.29). Thus (4.10) [Zll
with ~0o = A l z l ;
(pl = _ g o .
(4.11)
Exact Controllability of the Wave Equation with Neumann Boundary Control
279
But by (4.6), (4.10), and (4.5) we obtain
-IIA~,(t, o, ~,)11~2(o,~,,2~ro ) =
I
~o +~o, d ~ , . 1
2
2
(4.12)
Moreover by (4.11)
II{Zo, z, } IIL2(.~ xt~(a"2.' = II{-- ~ ', A~P°} IIL2(.)~ l~ (A"~1' = II{~°, --~l}[[~(a'/~)×L~m).
(4.13)
Using (4.12)-(4.13), we see that inequality (4.3) becomes precisely inequality (4.1) with Cr = C ] .
4.2. The Case Fo = The counterpart of Theorem 4.1 is now:
Problem (1.1) is exactly controllable on the space L2(fl) x [H~(I~)] ' over the time interval [0, T], 0 < T < oo, by means of controls v c [H~(0, T; L2(F))] ' if and only if the following inequality holds for the solution ~ of problem (3.2) with
Theorem 4.2.
ro = ~ :
[for all o , ~pl~ Hl(f~) x L2(I']) for which the left-hand side of k the above inequality is finite, where Cr is a positive constant depending on T but not on o , ~ ~.
(4.14)
Proof of Theorem 4.2. Let v be as before. The operator ~ r is now (see (3.24)): (A+ I) f ~ S ( T - t)NlV(t) dtl,
yJ; l
(4.15)
A+I,I°
the counterpart of (4.2). The exact controllability in question is equivalent to the ontoness property ~r:
[Hi(0, T; L2(F))]'~ ~ ( ~ r )
onto
) L2(f~) x [H~(I))] '
(4.16)
or equivalently
I[ I°lll, ~*~
t,,,(o,~,~=(~.j,-> C~ll{=o, z,}ll~=(.~×t.'(.)~,
for some
CT>O and all [Zo, z~]~ ~ ( ~ * ) .
(4.17)
Using, this time, ~ ( ( A + I) 1/2) = H i ( f l) (cf. 2.5c), the same norms, and proceeding
280
I. Lasiecka and R. Triggiani
as in Theorem 4.1, we likewise obtain (4.12), with ~ 0 = ( A "[- I ) - ' Z l ,
1 = -Zo,
(4.18)
the counterpart of (4.11). Moreover, by (4.18) and (2.5c) II{zo, z,}ll
=
(A+
= I1{ °,
(4.19)
the counterpart of (4.13). Then, by virtue of (4.12) and (4.18)-(4.19), we see that inequality (4.14) becomes precisely inequality (4.14) with Cr = C~. Theorem 4.2 is proved. [] 4.3. Completion of the Proof of Theorem 1.3 Let Fo # 0 . Then Theorem 1.3 follows by simply combining Theorem 4.1 with Theorem 2.5. If, instead, Fo = Q, then Theorems 4.2 and 2.6 combined produce Theorem 1.3. [] 4.4.
Proof of the Corollary to Theorems 1.2 and 1.3
We apply the interpolation theorem [LM, Theorem 5.1, p. 27] to the operator 5f* which, under the present assumptions, satisfies LP*: continuous ~ ( A u2) x L2(12) ~ Lz(E1)
(4.20)
by (3.10), (3.27) and Section 3, and Z¢*: continuous L2(f~) x [H~o(f~)]'-~ [H~(0, T; L2(F1))] '
(4.21)
by (4.3) and (4.17) above. Hence ~ * is continuous between 1 r [@(A '/2) >
~(A(1-°)/2) x
0 t [Hro(~)]
and [H°(0, T; L2(FI)), [n~(0, T;
g2(rl))]']0
=
[H°(0, T; L2(F1))] '
as it follows via the duality theorem on interpolation [LM, Theorem 6.2, p. 29]. This then means that Let is onto in the opposite direction. The prooof of the corollary is complete. []
5.
5.1.
Exact Controllability on L2(f~)× [H1(I~)] ' in the Neumann Case (Fo = Q) in the Absence of Geometrical Conditions on fL
Equivalence to an A Priori Inequality. Ontoness Approach
We now establish exact controllability of problem (1.1) in the Neumann case (Fo = Q) on the space L2(fl) x [H1(~)] ' without requiring geometrical conditions on fl. We shall see that in order to achieve this goal, a larger class of controls v
Exact Controllabilityof the Wave Equation with Neumann BoundaryControl
281
in (1.1d) (with Fo = O) is needed, which moreover possesses a special structure. This class is introduced as follows. Given fl, we fix a point Xo in R" and set F(x °) = {x ~ F: (x - x°) • v(x) >-0},
(5.1a)
r,(x °) = r\r(x °)
(5. lb)
as in (1.14). Then the class of controls v in (1.1d) with Fo = O is defined by O C
v =
[HI(0, T; L2(F(x°)))] ',
(5.2)
v2 ~ L2(o, T; [ n l ( r . ( x ° ) ) ] ' ) .
We shall henceforth set in this section the following notation: L2(Z(x°)) - L2(0, T; L2(r(x°))); Lz(Z,(x°)) = L2(0, T; L~(r,(x°))).
(5.3)
In the main statement below, we need the concept of tangential gradient V~q, for a function ~b~ C1(~) on F (or part thereof; in our case below: F.(x°)). At each point of F (sufficiently smooth) consider the unit outward normal v and a, say, orthogonal system of unit vectors zl,. •., ~',-1 on the tangent plane. We have ,-1 0~ ,-1 0@ V~ = (V~" v ) v + i=1 ~ (V~'ri)ri=--V+ov i~=1~ r,.
(5.4a)
Let, in addition, O@/Ov = 0 on F (or part thereof), so that
4,, = ~,4,, cr~= first-order tangential operator on F.
(5.4b)
Then we define n--1
IIIv@l libel, = Z I~,¢,F- IIIv~l tlb,r~. i=1
(5.5)
The main result of this section is:
Theorem 5.1. Problem (1.1) with F o = O is exactly controllable on the space L2(fl) x [HI(ll)] ' over the time interval [0, T], 0 < T < ~ , by means of controls v given by (5.2) /f and only if the following inequality holds for the solution ~p of problem (3.2) with Fo = •:
(x°)
~, dr(x °) dt+
I v ~ l 2 dr,(x °) at
do dr,(x °)
+ f o r l r ~2 d~;>- C~ll{~°,
~
' }It.'<.~×L2~.)
(5.6)
for all {q~o q~x}c Hi(I-I) × L2(fl) for which the left-hand side is finite, where Cr is a positive constant depending on T but not on tp°, tp1. Proof of Theorem 5.1. (Modification of proof of Theorem 1.3.) Step O. We introduce the operator /~/1: L2(F(x°))x[Hl(F.(x°))]'->nl/2(D)
282
I. Lasiecka and R. Triggiani
defined by '(h-1)w=0
Ow
w= ]Q [gll ¢:~ Ov = gl
onlY, on l~x°j,
(5.7)
11g21
OW
,0-~= g2
on r,(x°).
Let NI* be the adjoint of ]Q1-"
(5.8) ] L2(F(x°))×[HI(F,(x°))] '
Lemma 5.2. Forfe ~(A) (cf (2.1) with ro= Q), we have
5I*(A+ I)f= ([~r*(A + l)f]~, [/(/*(A + I)f]2) e U ( r ( x ° ) ) x [Hl(F,(x°))] ' and [ 5/1"(A +
I)f]l =flr(x o) ~ L:(r(x°)),
(5.9a)
([/Q1*(A+ I)f]2, g2)ts'(r,(x°))] ' = ( f g2)L~(r,(x°)),
Vg2~[Hl(F.(x°))] ', (5.9b)
[/Q1*(A+ I)f]2 = A~flr,(x °) ~ [H~(r,(x°))]
',
(5.9c)
where Ab (b = boundary) is a first-order tangential operator on F,(x °) (with smooth coefficients) which defines an isomorphism Ab: H~(F,(x°))--> HS-l(F,(x°)), with bounded inverse Ab 1, self-adjoint o n L 2, i.e., Ab = A* where (AbUl, U2)L2(F,(x°))=(U 1, A*ub~)L~(r,(x°,,
(5.1o)
u~ e L2(r,(x°)).
Proof of Lemma 5.2. With [gl, g2] ~ L2(F(x°)) x [HI(F.(x°))] ', we proceed as in the proof of Lemma 3.5 and find by the Green second theorem and (5.7) -(]Q*l(a+l)f,
:12 )L2(r(xO))x[H,(r.(xO))],= -((a-be)f, JQ1:12 )L2(a)
g2 / L2(a)
= -(f,, gl)L~(r(x°))- (f g2)t.2W,(~°)) from which (5.9a, b) follow at once. Using (u~, u2)tH,¢r.(xo))l,= (A-lul, A-~u2)L2(r.(xo)), in (5.9b) yields (5.9c) by (5.10).
u, ~ [ H l ( F , ( x ° ) ) ]
'
(5.11)
[]
Exact Controllability of the Wave Equation with Neumann Boundary Control
Step I.
283
In place of (4.15) we now have Io
yt(T)
-
dt,
._-[Vl(t)
(A+I) Io C(T-t)lVl[Vl(t)[ dtv2(t)
(5.12)
~(~eT) = {[v,, v:] ~ [H'(0, T; L:(r(x°)))]' x U(0, T; [n~(r,(x°))] ') x [y(T), y,(T)] ~ L2(fl) x [n~(f~)]'}, Exact controllability in question is equivalent to ~r:
~(~T)
onto
, L2(I~)X [nl(f~)] ',
(5.13)
which in turn is equivalent to
for some
C~->O
[zo,Zl]~ ~ ( ~ * ) .
and all
(5.14) I
If [ ~ * zo ] , i=1,2, denote the two component of ~ . [ z o on Y~-= IZl Z1 -1i k [HI(0, T; L2(F(x°)))] ' and Y2----L2(0, T; [H~(F,(x°))]'), respectively, we then have zl
Y~xV2
21
I Y1
I.
Zl .12 Y2
We now set (as in the proof of Theorem 4.1) ~o(t, tp°, ¢p~)= C(t-
T)~°+S(t- T)cp1
(5.16)
for the solution of problem (3.2) with
¢°=(A+l)-lz,,
¢'=-Zo.
(5.17)
Now let A be the isomorphism Hi(0, T; L2(F(x°))) onto L2(E(x°)), see (5.3), self-adjoint on LE(y~(x°)),which satisfies the same properties (4.4)-(4.6) with F1 there now replaced by F(x°). We obtain from (5.12)
I.
(,,, ,,
IZllJl/v~
\
I
17"11 2 Y2
L
=oi
I zl 1.1",/L2(O,T;[.l(r.(.°));l')
(5.18)
On the other hand, by proceeding as in the proof of Theorem 4.1, equations (4.3)-(4.10), we likewise find, using now H1(12)= ~ ( ( A + i)1/2) and (5.9a),
Thus, by (5.9a), if we compare (5.18) with (5.19) we obtain
the counterpart of (4.10). Similarly, by (5.9c),
Thus, using the properties (4.4)-(4.6) for the present A we find, from (5.20),
the counterpart of (4.12). Also, from (5.21), by the property of Ab
Thus, combining (5.15) with (5.21)-(5.22) we obtain
Moreover, as in (4.19), we have, by (5.17), Since Ab is a first-order tangential operator, then (5.14), (5.24), and (5.25) yield (5.6), as desired. []
It suffices to combine Theorem 5.1 with Theorem 2.8.
Acknowledgment The authors would like to thank J. L. Lions for a stimulating exchange of correspondence between February and July 1986 during which most of the results of this paper were obtained.
Exact Controllabilityof the Wave Equationwith NeumannBoundaryControl
Appendix A.
285
Sketch of Proof of (2.18)
The identity div(~h)= h. V ~ + ~bdiv h and the divergence theorem give
Inh. V ~bdt~ = Ir~bh. v dF - Ia~b div h dfL
(A.1)
Multiply both sides of (2.12a) by h.V~ and integrate in Q. As to the left-hand side, we integrate by parts in t, use ~,h'V~t =½h'V(~2), and identity (A.1) with = 2 . We obtain
fff,,h.V¢dO=f[¢,h.V~,][af~-½f~,h.vd2+ifo~,~divhdQ.
(A.2)
As to the right-hand side, we use Green's first theorem, the identity Vw" V(h" Vw) -- HVw" Vw +½h"v(Iv wl2) and identity (A.1) with ~b= IV¢]2. We obtain A~o(h.V~o) dQ =
ov(h.V~o) d ~ -
HV~.V~odQ
l fJv~12h'vd~,+½folv~12divhdQ.
(A.3)
Equating the left-hand side (A.2) with the right-hand side (A.3) results in (2.18).
Appendix B.
The Minimal Norm Steering Control
Once exact controllability is established, the following elementary argument provides the minimal norm steering control u °. We first carry out the reasoning for an abstract equation, and then specialize its conclusions as they apply to, say, problem (1.1) with Fo = Q, on the state space Hl(f~)x L2(I~) with control space L2(•).
Abstract Treatment. .~ = My + Nu,
Consider the abstract equation
y(O) = Yo,
(B.1)
M being the generator of a strongly continuous semigroup of operators on the Hilbert space Y and N: U ~ ( N ) o Y being a linear, generally unbounded operator from another Hilbert space U to Y, with M-1N continuous from U to Y (without loss of generality we may assume that M is boundedly invertible). The solution to (B.1) with Yo= 0 is ~T u : M
Io e~C(T-t)M-l~u(t)
dt.
(B.2)
286
I. Lasiecka and R. Triggiani
Let z be a target state in Y and consider the following minimization problem: minimize J(u)
1 u 2 =~JI IIL2~o,T;u~
over all u ~ L2(O, T; U) such that ~TU =Z, under the (exact controllability) assumption that there exists at least one such u. Let us indicate by ((,)) the duality pairing between Y' and Y with respect to a (pivot) Hilbert space X satisfying either X c y (as in the specialization below) or else X c y'. Then the Lagrangean can be written as
L(u,p)=½(U,U)L2(O,T;U)--((p,~TU--Z)),
p c r'.
Taking L, = 0 yields uo= ~ p O ;
thus
z = ~TU ° = ~ r ~ - p °,
(B.3)
where ~ is the conjugate operator from Y' to L2(0, T; U) defined by ((v, ~ r u ) ) = ( ~ - v , U)L2(O,T;O), V ~ Y'. From (B.2) it readily follows that ~pO=
~ . ea*(T-Oj-lpO,
(B.4)
where J is a norm-preserving isomorphism Y onto Y'. Note that ~e*~= ~ J
(8.5)
is the Hilbert space adjoint Y ~ L2(0, T; U) of ~T which we have used in this paper. Moreover, from (B.3) and (B.5) we obtain
( ( p ° , z ) ) = ( ( p ° , ~ r ~ c P ° ) ) = II~p°ll ~=~o.~;u~ = I1~1" - 1 p 0 ii~~- CT II/-~p°tl ~
= CT IIp°ll ~,,
(B.6)
where we have used the lower bound inequality for 5f* which states that ~ r is L2(0, T; U) onto Y, i.e., the exact controllability assumption on (B.1). Thus, the operator ~r&V~- defines an isomorphism Y' onto Y by the Lax-Milgram theorem applied to (B.6), and from (B.3) we have
pO = [pO, pO] = ( ~ T ~ ) - ~ Z ~ y,.
(B.7)
Hence, by (B.3), (8.4), and (B.7) we find that the optimal minimal norm steering control is given by U0 = ~ - ( ~ T . ~ P , ~-)-lz = ~ *
e~*(T-t)J-l(~.~T,~-)--lz
(8.8)
in terms of the target z and the dynamics M, 93, 5~T.
Specialization to the Hyperbolic Problem (1.1) with U = L2(E), Y = H'*(f~) x L2(~'~), and Fo = Q. Here we take X - - L2(I~) x L2(I~) so that Y' = [Hl(f~)] ' x LE(f~). Using the operator model for problem (1.1) which was introduced and used in [LT3], [T2], [T3], and [FLT, Appendix B], we readily obtain that (B.4) and (B.8) specialize to uO= ~ . p O = ~ . ea*(T-,)j-lpO= --~]x~ L2(•). (8.9)
Exact Controllabilityof the Wave Equation with Neumann BoundaryControl
287
Here, pOe y, is given by (B.7) and M is the realization o f - A + I with homogeneous Neumann boundary conditions in L2(fl), so that ~(M~/2)= HI(O) with equal norm as in (2.5c). Moreover, the isomorphism J from Y onto Y' is defined by j = M~/2 x / , where I is the identity map on L2(I)). Finally, we have that ¢ solves the following homogeneous problem: ~tt = A ~ ,
q~°= p°2,
~ox= Mp °,
(B.10)
0_~ ~= 0,
see, e.g., [T2] and [FLT, Appendix B, equations (B.10)-(B.13) with a = 1].
Remark B.1. An argument very similar to the one above gives the following result. Let Zo~ Y be an arbitrary initial point. The optimal minimal L2(0, T; U)norm control, which steers Zo to the origin at t = T along the dynamics (B.1), is given by uO ~
~
~ --1
--~T(~TZPT)
(BAD
eaTzo
under the (null controllability) assumption that there exists at least one steering control. If, moreover, M = - M * so that e ~t is a unitary group on Y, then we readily obtain the identity #, # -1 e ~ T ~* ~ ~~ -1 ~T[¢.~.T¢aCffT) -- o,(ffT(~.~T~ffT) ,
(B.12)
where we have set ~ T = e-~CT~T = e a * T ~ T : L2(0, T; U ) ~ r
(a.13)
so that ff~- = ~ J e - ~ * T J
-~ = ~*T e - ~ * T J - ' = ~ * e a T J - ' : Y ' ~ L2(0, T; U).
(B.14)
Thus, in the case M = -M*, using (B.10), (B.11), (B.5), (B.13), and (B.4), we see that the optimal (minimal norm) steering control is given by nO
"~ ~ "4~" --1 :~ --..c~*T --1 " "4~ --~T(~T~,T) go=--~PTJe J (~To~T)
--1
gO
"~--- - ~ * e ~*(T-t) e-'Sd* T T; ~- ,-j1[,:~T,:~T)r~4~x--1 Z0
=-~*
e~tj-l(-~r~-)-lZo=-~* e'~"(~r~-Y)-'Zo.
(B.15)
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I. Lasieckaand R. Triggiani
This control is the one that J. L. Lions uses in his method, on which we shall have more to say in Remark B.2 below. Remark B.2.
In this paper we have preassigned the space V:~ of controls on [0, T], as well as the target space Z where we seek exact controllability. Various choices of the pair { V:~,, Z} were considered, and others may be studied by similar techniques. In any case, the technical crux consists in establishing the inequality
II zllv
c llzllz,
z,
(B.16)
for some suitable positive constant cr, independent of z, where ~ * is the Hilbert adjoint Z ~ ~(ZP*)~ V~ (Z is pivot space). Alternatively, having chosen V~,, we may seek the largest space where exact controllability is attained for the dynamics defined by the operator &Pr in (B.2). To this end, we proceed as follows. Let Y be a Hilbert space such that ~7- has (i) domain ~(Z#r) dense in Vx, and (ii) range R ( ~ r ) dense in Y so that the Hilbert adjoint Z#*: Y D ~(Z#*) -> V~ (Y is pivot space) is injective: ~*y=0,
y~(~*)
~
y=0.
(Ba7)
This is the property of approximate controllability of Zfr in Y. Then, by the Lax-Milgram lemma we have that (i) 5qTZf* is an isomorphism from all of ~(&e*) onto [~(&f*)]'
(B.18)
where [ ]' denotes duality with respect to the pivot space Y; (ii) ~*T(3?r~*) -1 maps all of [ ~ ( ~ * ) ] ' into Vz,.
(B.19)
Thus, setting for all z ~ [ ~ ( ~ * ) ] ' u° =
1
(B.20)
we see (as in going from (B.2) to (B.8)) that u ° in (B.20) is the minimum V:~ -norm control which steers the origin at time t = 0 to the target point z at time t = T along the dynamics (B.1) or (B.2). Moreover, we may verify that the space [ ~ ( ~ * ) ] ' does not depend on the particular choice of the space Y subject to the condition that ~ r has both domain dense in V~, and range dense in Y. The space [~(Zf*)]' coincides precisely with the totality of points to which the origin can be steered on [0, T] by means of V~,-controls along the dynamics (B.1), which for time reversible dynamics as (1.1) is the space of exact (null) controllability. Thus [ ~ ( ~ * ) ] ' coincides with the space F' in J. L. Lions's notation.
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289
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Accepted 3 February 1988