Appl Math Optim 18:241-277 (1988)

Applied Mathematics and Optimization O 1988Springer-VerlagNew York Inc.

Exact Boundary Controllability on L2(i~)x H - l ( ~ ) of the Wave Equation with Dirichlet Boundary Control Acting on a Portion of the Boundary 0i~, and Related Problems* R. Triggiani Department of Applied Mathematics, Thornton Hall, University of Virginia, Charlottesville, VA 22903, USA

Abstract.

C o n s i d e r the wave e q u a t i o n d e f i n e d on a s m o o t h b o u n d e d d o m a i n ~ c R" with b o u n d a r y F = F 0 w F 1 . T h e c o n t r o l action is e x e r c i s e d in the D i r i c h l e t b o u n d a r y c o n d i t i o n s o n l y on F1 a n d is o f class L2(O, T: L2(F1)); instead, h o m o g e n e o u s b o u n d a r y c o n d i t i o n s o f D i r i c h l e t (or N e u m a n n ) t y p e are i m p o s e d on the c o m p l e m e n t a r y p a r t Fo. The m a i n result o f the p a p e r i s a t h e o r e m which, u n d e r g e n e r a l c o n d i t i o n s on the triplet {1), Fo, F1} with F o # •, g u a r a n t e e s exact c o n t r o l l a b i l i t y on the space L2(I)) x H - l ( l ) ) o f m a x i m a l r e g u l a r i t y for T greater t h a n a c o m p u t a b l e time To> 0, w h i c h d e p e n d s on the triplet. This t h e o r e m g e n e r a l i z e s p r i o r results b y L a s i e c k a a n d the a u t h o r [ L - T . 3 ] ( o b t a i n e d via u n i f o r m s t a b i l i z a t i o n ) a n d b y Lions [L.5], [L.6] ( o b t a i n e d by a direct a p p r o a c h , different from the o n e f o l l o w e d here). The key t e c h n i c a l issue is a l o w e r b o u n d on the L 2 ( Y 0 - n o r m o f the n o r m a l derivative o f the s o l u t i o n to the c o r r e s p o n d i n g h o m o g e n e o u s p r o b l e m , which e x t e n d s to a larger class o f triplets {1), Fo, F1} p r i o r results b y L a s i e c k a a n d the a u t h o r [L-T.3] a n d b y H o [H.1]. * This research was partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365. This paper was presented at the IFIP WG7.2 Conference on Boundary Control and Boundary Variations held at the D6partment de Math6matiques, Universi'te de Nice, 10-13 June 1986; at the International Conference on Control of Distributed Parameter Systems held at Vorau (Austria), 6-12 July 1986; and at the Second Workshop on Control of Systems Governed by Partial Differential Equations held at Val David, Quebec, 5-9 October 1986.

242

1.

R. Triggiani

Introduction; Literature and Motivation; Statement of Main Results

1.1. Introduction Throughout this paper 12 is an open, bounded domain in R n, with sufficiently smooth boundary 012--F. In 12 we consider the following nonhomogeneous Dirichlet problem for the wave equation in the solution w(t, x): fw, =Aw J w ( 0 , . ) = w °, /W]~o=O I,wk,=-u

w,(0,.)=w'

in (0, T)x12--- Q, inlI, in (0, T)Xro-=5;o, in (0, T) xF1---~I,

(1.1a) (1.1b) (1.1c) (1.1d)

with control function u ~ L2(0, T; L2(F1)i- L2(E1). Here, F0 and F 1 a r e subsets of F, with Fow F~ = F, Fo possibly empty Q, while F1 is nonempty and relatively open. We also set (0, T ) × F-=E. In qualitative terms, the problem of exact boundary controllability for (1.1) is as follows. Given the triplet {1), Fo, F~} we ask whether there is some To > 0 (depending on the geometry of the triplet) such that if T > To, then the following steering property of (1.1) holds true: for all initial data w°, W 1 in some preassigned space Z = Z1 x Z2, there exists a suitable control function u~L2(O, T; L2(F1)) whose corresponding solution of (1.1) satisfies w(T, • ) - w,(T, • ) = 0. If this is the case, we say more precisely that the dynamics (1.1) is exactly controllable in the space Z over the time interval [0, T ] 1 (be means of L2-controls). In Section 5 we also consider the case where (1.1c) is replaced by Ow/Ov]~o=- O.

1.2.

Literature and Motivation

There exists a large body of literature regarding exact controllability problems for the wave equation (or other hyperbolic dynamics) with control action either in the Dirichlet boundary control (B.C.) as in (1.1c-d), or else in the Neumann B.C. (or variations thereof). Among the numerous works on the nonhomogeneous Dirichlet case prior to 1985, we cite only a few: Russell [R.2], [R.3], Littman [L.7], [L.8], Lagnese [L.1], [L.2], Ralston [R.1], and others, and we refer to the other works quoted in these papers. It is important to observe, however, that in all of the literature prior to 1985, it was always assumed that the space Z of arbitrary initial data w °, w 1 be "smooth" in one way or another; more precisely, it was assumed that (w °, w 1) c Z -= H2(f~) x H1(12) (1.2a) for general domains D [R.2], [L.2] with Fo = O or

(W°, W1) e Z ~ H i ( D ) × L2(l)) (1.2b) under appropriate geometric restrictions on 12 [R.3], [L.1], [L.8] 1Steeringany initial condition (i.c.) in Z to rest (0, 0) in [0, T] is equivalent--by time reversibility of (1.1)--to steering the origin (0, 0) to any (target) final condition (f.c.) in Z in [0, T]; hence, to steering any i.c. in Z to any f.c. in Z over [0, T], by L2(El)-controls.

Exact Boundary Controllabilityo n L2.(aQ) × H-l(Ft)

243

(Littman's interesting treatment in [L.7]--valid for hyperbolic and parabolic partial differential equations with constant coefficients--assumes C~(f~)-initial data). We shall substantiate below that the above spaces Z are not the natural state space for (1.1); as a consequence, results of existing literature on exact controllability for (1.1) on the space Z given by either (1.2a-b) or C°~(I~) x C°(l)) are inadequate when it comes to the study of the regulator problem associated to (1.1) [L-T.4]. More precisely, we claim that the natural state space for (1.1) is the space L2(f~)x H - I ( I ) ) . This follows from the following (optimal) regularity theorem.

Theorem A.

For all 0 < T < oo we have, for the unique solution of (1.1),

(W°, w1)EL2(I-~)xH-I([~) u e L2(£)

~wE C([0, ~

T]; L2(O)), [w, e C([0, T]; H - ' ( f l ) ) ,

(1.3)

continuously.

Remark 1.1. Theorem A was first proved in 1982 in [L-T.2]--improving upon the result [w(t), w,(t)] ~ L2(0, T; L2(I)) × H-~(I))) obtained in [L-T.1]--and, independently, and at about the same time, in [L.3]. The treatment in [L-T.1] and [L-T.2] also makes use of an abstract input ~ solution operator model for problem (1.1), see (2.1) below (introduced in [T.1] and valid for a general time-independent elliptic operator in place of -A). A comprehensive treatment which restudies the case u 6 L2(£) with notable technical simplifications over [L-T.1] and [L-T.2] and includes also the cases of u smoother than, or less smooth than, L2(£) is given in [L-L-T.1]. We shall see below that the technique employed in proving the main exact controllability theorem in this paper is the same as the multiplier technique of [L.4] and [ L-L-T. 1] used to prove the regularity of the normal derivative of the homogeneous problem (1.1) with u -= 0; see more detailed comment in Remark 2.1. Theorem A by itself justifies the claim that exact boundary controllability for (1.1) should be studied in the natural state space Z = - L 2 ( D ) x H ~(~). A further confirmation--and motivation--comes from the study of a consistent optimal quadratic cost theory for the hyperbolic dynamics (1.1) in the associated regulator problem [L-T.4]. According to Theorem A, a correct setting of the regulator problem for (1.1) is: Minimize the quadratic functional

J(u, w)~ I5 I1w(t)l]22~m + l[w'(t)ll2-'m)+ over all u ~ L2(0, oe; L2(F)).

I[u(t)l[22~r)

dt

(1.4)

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R. Triggiani

As is well known, before a meaningful study of the above regulator problem may be initiated, we are faced with the need to ascertain that the following finite cost condition (F.C.C.) holds true:

F.C.C. on L2(D.) x H-'(f~):

for each (w °, W 1) C L2(f~) x H-~(f~), there exists at least one fi c L2(0, oo; L2(F)), such that the corresponding solution ~ of (1.1) satisfies

( i

e L2 0, co; H_~(f~ ) ] , so

(1.5)

that the corresponding cost functional J is finite: J(fi, ~)
Theorem B. Let F o = Q (empty), F l = F . Consider problem (1.1) with u given in (dissipative) feedback form by

u

0 (A_lw,) ' Or,

u = outward unit normal,

(1.6)

where - A is the Laplacian A with homogeneous Dirichlet B.C. o n L2(~)). Then, if one sets [w(t), w,(t)]-= Sv(t)[w °, w l] for the resulting feedback system with (1.6) inserted in (l.ld), the following holds true: (i) Sv( t) defines an s.c. contraction semigroup on L2(f~) x H-l(f~) (feedback semigroup); (ii) (a/Ov)(A ~w,)~ L2(0, o0; L2(F)); (1.7) (iii) (strong stabilization)

SF(t)[wO, w~]--~O as t ~ c o ,

forall

(w°,wl)cL2(~)×H-l(~);

(iv) under certain geometrical conditions on ~ specified below--which in particular include the class of strictly convex domains f~--strong stabilization as in (iii) can be improved to uniform stabilization on L2(~)× H - l ( f l ) : there exist constants C, 8 > 0 such that

IIS~(t)lJ ~(L2(~)×H-'.~))~ Ce -~',

t>-O.

(1.8)

Exact Boundary Controllabilityo n L2(~Q.) × H-I(f~)

245

The geometric conditions on 1) are as follows: there exists a vector field h ( x ) = [ h i ( x ) , . . . , hn(x)]E C I ( ~ ) such that:

(a) h is parallel to p on F; (b) J ~ H ( x ) v ( x ) ' v ( x ) d l I > - p J ~ ] v ( x ) [ ~ , , d ~ , forall

forsome

p>0,

v ( x ) ~ L2(I~: R"),

where H ( x ) (transpose of the Jacobian matrix o f h) is given explicitly by (1.12) below.

The claim in the paragraph preceding Theorem B is now ascertained. First, by virtue of (1.6)-(1.7) and (1.8), uniform stabilization on L2(I~) x H-I(I~) plainly implies the F.C.C. (1.5) on the same space. Moreover, via the by now well-known "controllability via stabilizability principle" valid for time reversible systems-due to Russell [R.3]--Theorem B(iv) allowed [L-T.3] to establish, for the first time, the corollary that exact boundary controllability for (1.1) with F1 = F on the space Lz(fl) x H - I ( [ I ) over [0, T] does hold true, at least under the geometric conditions needed in Theorem B(iv), see [L-T.3]. A virtue of Russell's p r i n c i p l e .inherited in [L-T.3]--is that it provides a constructive way to define a boundary control u which steers the initial condition (w °, w ~) to rest (0, 0) over [0, T]. This control is expressed, in fact, in terms of the stabilizing feedback of the closed-loop system, in our case (O/Ou)(A-~wt). It is therefore natural to ask whether the geometric conditions on D under which exact controllability for (1.1) (with F1 = F) on the space L2(I))x H-l(l~) was obtained constructively in [L-T.3] as a consequence of the uniform stabilization result (1.8) are "intrinsic" to the problem, or rather due to the special choice of the stabilizing feedback (1.6). A complete answer to this question was provided a year later by Lions at the IFIP Workshop held at the University of Florida, 2-6 February 1986 (private communication, followed by [L.5]). Lions's definitive result is that exact controllability on L2(f~) × H-a(l)) for problem (1.1) with F 0 = Q and F1 = F holds true for any domain 1) (with 0 l ) = F sufficiently smooth). Geometrical assumptions on f~ are imposed, by contrast, when Fo # • and thus the control u is active only in the portion F~ = 0f~\Fo of the boundary 01) = F.

Theorem C [L.5; L.6]. (i)

Let F o = Q and so F1=F. There exists T o > 0 (depending on 1)) such that for T > To, then, for any (w °, w 1) c L2(II) x H ~(1~), there exists u ~ L2(~,) such that the corresponding solution of (1.1) satisfies w( T, O) = w,(T, " ) = 0 . (ii) 2 Let Fo ~ Q. I f there exists a point Xo ~ R n such that ( X - X o ) • u(x)<-O

for

X~Fo,

t,= outward unit normal,

(1.9)

then the same conclusion as in part (i) holds for problem (1.1a-d), with suitable control u c L2(E1).

2 Part (ii) is actually not stated in [L.5] and [L.6]. It was part of the private, oral communication given by Lions at the 1FIP Workshop in February 1986.

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R. Triggiani

Lions's p r o o f of his T h e o r e m C is based on a general method, which we shall call the (F, F', A ) - m e t h o d , and for which we refer to [L.5] and [L.6]. 3 Unlike the p r o o f o f exact controllability given in [L-T.3] as a consequence o f (i) uniform stabilization in F '-= L2(f~) x H - l ( f ~ ) u n d e r geometrical conditions in 12 and (ii) Russell's principle, Lions's p r o o f o f the more general T h e o r e m C is only in principle constructive, for it requires the inversion A -1 o f the isomorphism A ~ ZP(F, F'). (For numerical work in this direction see [G.1].) The assertion that A is an isomorphism is b a s e d in turn, on L e m m a 2.2 below, in the special case Fo = Q (where a radial vector field h(x)= x - X o suffices), a b o u t which we shall have to say more in R e m a r k 2.1.

1.3. Statement of Main Result The main p u r p o s e o f this paper is to provide an alternative a p p r o a c h to the problem o f exact controllability o f (1.1), which recovers T h e o r e m C(i) and extends T h e o r e m C(ii) o f Lions. Moreover, classes o f domains 12 will be given in the illustrative Section 3 below, where our generalized version o f Theorem C(ii) will provide a more precise and refined information on the active portion F1 of the" b o u n d a r y 0f~, than it is possible to obtain by applying the sufficient condition (1.9) based on radial vector fields x -Xo. M o r e precisely, by resorting to our main T h e o r e m 1.1 below, we obtain an active part F1 of the b o u n d a r y strictly smaller than it is possible to achieve by use of condition (1.9). This is so, since we shall allow more general vector fields than radial vector fields. O u r p r o o f here is not constructive either, but it does provide a new insight into the problem. 4 Moreover, once exact controllability is established, the elementary argument o f A p p e n d i x B furnishes the minimal n o r m control. Such minimizing control is precisely the one provided by Lions's method.

Consider problem (1.1), with Fo possibly nonempty. Assume that there exists a vector field h ( x ) = [ h l ( x ) , . . . , h n ( x ) ] 6 C 2 ( ~ ) such that the following three conditions hold: Main Theorem 1.1.

(i) h(x). v(x)<-O

for x 6 F o ,

v=outwardunitnormal.

(1.10)

(ii) f n H ( x ) v ( x ) . v(x) d12>--p falv(x)12R,, d~2

for some constant p > 0 ,

for all v ~ L2(12; R"),

(1.11)

3 For problem (1.1), we have F = L2(~) x H~(~), so that F'= L2(~'). ) × n 1(~), the space where exact controllability is achieved. 4 This is even more evident in the case of control acting in the Neumann B. C. over all of 0fl, or else on the portion F~ of O~, while homogeneous Dirichlet B.C. are imposed on Fo = 0f~\F~, as it appears from work in progress by Lasiecka and the present author [L-T.5].

Exact Boundary Controllability o n L2(D.) x H-I(I~)

247

where I °h__2 . . . °h__2 c3Xl OXn H(x) =0 h.___~

0 h_.__~

Ox~

Ox,

(1.12)

(A sufficient, checkable condition for (1.11) to hold is that H(x)+ H*(x) he uniformly positive definite on •.) (iii)

p >

2Gh~,/--~v

(this may be removed, see Remark 4.3),

(1.13)

where

f 4J2da<-cpfrv~l2da,

4Gh=--m-ax[V(divh)l;,

q,~Hl(a) (1.14)

( Cp = "Poincard' s constant"). Then there is To>0 (to be specified below) such that if T> To, then, for each (w °, w l) c L2(I)) x H-I(I)), there exists some u c L2( E1) such that the corresponding solution of (1.1) satisfies w(T) = w , ( T ) = 0 ,

wW] c C ( [0, T]; H-'(I~) L2(I)) '~ ]"

(1.15)

For To we can take To -

2Kh +4"-~p, p --2GhX/~p

Kh = D j - ~ +

M~ ;

(1.16)

2D~ --- m axldiv hi;

Mh ------m~xlhl.

(1.17)

1.2. If we specialize h(x) to be a radial field h (x) = x - Xo, xo is a fixed point in R", then H(x)== identity and

Remark

div h =- n = d i m l ) ;

Gh=0,

n

Kh = ~ 4-~ + m~xlx- xol

(1.18)

and assumptions (1.11) and (1.13) are automatically satisfied (with p = 1). Thus, with Fo nonempty, Theorem 1.1 applies, in particular, if there exists a point Xo~ R" such that

(X-Xo)'~,(x)<-O

for

x~Fo.

In this case, using (1.16) and (1.17), we see that we can take To to be To = (n + 1)v/-~p + 2 m~xlx-Xol.

(1.19)

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R. Triggiani

If, instead, Fo is empty, Theorem 1.1 applies with no conditions at all on Ut (except for some smoothness of OUt), thereby reproducing (through a different proof) Lions's Theorem C. In this case the smallest value of maxc~lx-xol is achieved by selecting Xo to be the center of the smallest sphere containing I), so that, if rxo denotes the radius of this sphere, then maxr~[x- Xo[ = r~o. Regarding the regulator problem for the dynamics (1.1) with F~ = F (indeed, with a general elliptic operator in place of -A) a rather complete theory was developed in [L-T.4], which does not require any assumptions of smoothness on the "observation" operators, and thus it applies to the cost functional (1.4). It made the assumption, however, that the F.C.C. (1.5) holds true, an assumption known at that time to be nonempty on the strength of the uniform stabilization/exact controllability result in L2(~ ) × H-I(Ut) of [L-T.3]. Now that exact controllability for problem (1.1) with F1 -- F has been established for any bounded domain Ut (with sufficiently smooth 0II), we can conclude from Theorem C, [L.5], [L.6], and [L-T.4]: Theorem 1.2.

The regulator theory of [L-T.4] with cost functional (1.4) applies to problem (1.1) with F1 = F with no assumptions on fl (except for sufficiently smooth F = OUt). It provides, among other things, a boundary feedback operator u ° : - B * P [ w ° ( t), w°(t)] (in the notation of [L-T.4], where [u °, {w°, w°}] is the unique optimal pair of problem (1.4) and P is a Riccati operator) which produces exponential decay in the uniform norm of L2(['~ ) >(H-I(~'~), thus reversing the implication of Russell's principle. Thus, exact controllability and uniform stabilization o n L2([-~) X H l(ut) (or the EC.C. (1.5)) are all equivalent properties for the dynamics (1.1) with F1 = F, and they all hold true for any bounded domain D (with sufficiently smooth F = OUt).

Remark 1.3. Theorem 1.1, and hence Theorem 1.2, can be extended to the generalized wave equation with A replaced by V(~¢V) where ~4 is a symmetric constant matrix. For the variable coefficient case, we refer to Section 4. Remark 1.4 (on the triplet {1-/, F0, F1}). The proofs in Sections 2, 4, and 5 below require the existence of a dense set of initial data of the homogeneous problems ((2.12), (4.4), and (5.12), respectively), whose corresponding solutions possess the regularity required to justify the calculations based on the multipliers below; in particular, the left-hand side of identities (2.18), hence (4.16), and (5.16). In the case Fon F1 = O, this requirement is fulfilled provided F is sufficiently smooth. This issue, however, is far more delicate in the case Fon F~ ¢ O and still not sufficiently investigated in the literature to make precise statements on {f~, Fo, F~}. In any case, to the extent that such a dense set of initial data exists, our results apply in principle also to the case FonF1 ¢ 0 , which in fact is crucial in Section 5: see footnote 6, p. 270.

Exact Boundary Controllability on

2.

Lz(II) x

H-1(12)

249

Proof of Theorem 1.1

Step 0 (Review of the author's previous results). The solution at time T to problem (1.1) with homogeneous initial data w° = w 1= 0 is given by the following input ~ solution operator ~ r : w(T, t = 0 ; w ° : 0 , w ' = 0 ) AIrS(T-t)l)u(t) w,(T,t=O;wO=O, wl=O ) = S f r u = A S ~ C ( T _ t ) I ~ u ( t )

,

(2.1)

see IT.l] and [L-T.1]-[L-T.4] for Fo = 0 . Here, /) is the Dirichlet map defined by y = 0 on ~, /gg = y ¢:~ =0 on Fo, (2.2) =g o n F1,

li

which is continuous L2(F1)---~L2(f~). Also, - A is the Laplacian with h6mogeneous Dirichlet B.C. on L2(f~), i.e., - A f = z~f @(-A) = H2(f~) n H~(f~); C(t) is the s.c. cosine operator generated by the self-adjoint operator - A on t L2( II ), t ~ R, and S ( t ) y = So C ( r )y dr, y e L2( f~ ). The following lemma--in the style of previous work [T.1]--will be needed. Lemma 2.0. I f £)* denotes the adjoint of l): continuous L2(f~) ~ L2(F1), we have for y ~ ~ ( A )

_l~.Ay=~Oy/Ov

t0Proof

onFl, o n Fo.

(2.3)

For y c ~ ( A ) and v c L2(F) we compute by Green's second theorem

-(19*Ay, v) L2(r) = - ( A y , £)v) t2(a) = (Ay,/)v) L2(a) = (y//)v))

L2(II)

+ (0~,/)v)

L2(F )

-(y,SDv)) \ ZOP

"

/ L2(F)

(by (2.2) and y ~ ~(A))

=(~' v)L2,v,,

[]

Step 1. The (regularity) Theorem A gives ~ r : continuous Lz(I~1)~ L2(12) x H -l(f~). By footnote 1, exact controllability of problem (1.1) on L2(O)x H-~(O) over [0, T] is equivalent to ~r:

L2(251)

onto

, L2(II) x H-~(f/).

(2.4)

This condition is equivalent, in turn, to IT-E1, p. 235] ~ * has continuous inverse: L2(II) x H - I ( I ) ) - , L2(Z~) ("continuous observability"), i.e., there is cr > 0 such that -> crll{z0, Zllll L2(~)×H-I(I]), Z1

L2(~)

(2.5)

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R. Triggiani

where Lf* is the Hilbert space adjoint, so that f o r f ~ L2(E0, y ~ L2(f~) x H - ' ( f l ) we have (Sfrf y)L:(n)×n-'(n)= ( f ~*Y)L~(~0.

(2.6)

Step 2. An equivalent partial differential equation characterization of inequality (2.5) is given by the following lemma. Lemma 2.1.

(i) For [Zo, zl] ~ L2([)~) X H-x([~), we have --O~,(t)

On~l,

(2.7)

where q~( t) is the solution of the following backward problem: in Q,

~tt = A~D

~Otlt=T~'~O1 inl~,

~t)lt=T = 0

~l~o-O

in So,

~o[s, ~ 0

in Yq,

(2.8)

where ~o° = A - l z l ,

1 = -Zo.

(2.9)

(ii) For any 0 < T < ~ , inequality (2.5) is equivalent to: there is C~ > 0 such that

, \U~/ d~l-> C)llko °, ~')II.~(~)×L=(.). Proof

(2.10)

(i) It is in the spirit of past work [L-T.2]. Since (with equivalent norms)

HI(f~) = @(A'/2);

H-I(f~) = [@(a'/2)]';

(y, Z)H-'(U)= (A-'y, Z)L2(a) (2.11)

we find as usual from (2.1), (2.11), and (2.2)

(Xo

= A =

S(r-t)Du(t)dt,

)

zo

+(f~

L2(FI)

C ( T - t ) L ) u ( t ) d t , z,) L2(n)

(u(t), I ) * A [ S ( T - t)Zo+ C ( T - t)A-1zl])L2(ro dt. 5

(2.12)

s In generaL, with A nonnecessarily self-adjoint, (2.12) holds with A, S(. ), and C(. ) replaced by A*, S*(.), and C*(.).

Exact Boundary Controllability on L2(~ ) x H - t ( ~ )

251

Thus, by (2.6) and (2.12), since C ( . ) is even and S ( . ) is odd,

(~*]:i)(t)=ff)*A[C(t-T)A-'z,+S(t-T)(-Zo)]

onE,.

(2.13)

But the solution of problem (2.8) is 9( t) = C( t - T ) 9 ° + S( t - T)91

(2.14)

and by (2.3) on

El

-~

( t) = D * A [ C( t - T ) 9 ° + S( t - T)91].

(2.15)

Comparing (2.13) with (2.15) and using (2.9) yields the desired conclusion (2.7) of part (i). Part (i) then immediately implies part (ii) via (2.9) and (2.11), so that 0

II{zo-- - 9 ~, z, -- 3 9 }11~=~>~--'~.~

is equivalent to

1

11{9°, 9 }]l nl(n~×L2(n~• (2.16)

Step 3. It remains to show if, or when, (2.10) holds true. The following lemma is the key technical issue of the exact controllability problem for (1.1).

Lemma 2.2.

Under the same assumptions in Theorem 1.1, there is To > O--which is given, in fact, by (1.16)--such that if T > To, then inequality (2.10) holds true with C ~ = Chlp(T-- To),

f , (a--~] 2 dX, >-ch..(Tkay/ >_ c , c , . . ( T -

c,.o -- p - 2 Q . / - ~ ,

To)II{IVg°I, 9')IIL~-~×L2~-~

(2.17a)

To)ll{9 °, 9'}11%o'.~×L=~.~,

(2.17b)

f lVg°l=dn>-c,llg°ll%~,

c,(l+Cp)=l, (2.17c)

where, by time reversal in (2.8), we can take 9 to be the solution of the homogeneous problem

9 , = A9 9(0,')=9°~H1(12): 9Ix--- 0 Remark 2.1.

9,(0,')=91~L2(E~),

in Q,

(2.18a)

in~,

(2.18b)

in Y-,.

(2.18c)

When Fo = Q (empty) and thus F1 = F, the opposite inequality of

252

R. Triggiani

(2.17a), i.e.,

\~/

dX<_CTII{~°, ~ }ll,~o'(.~×L2(m

(2.19)

always holds true (with 0f~ sufficiently smooth) for all 0 < T < ~ . This trace regularity result (2.19) for problem (2.18) was proved in [L-T.2, Theorem 2.1 and Remark 2.1, p. 279] (using an operator approach based on the abstract model (2.1) and the L2 regularity result of [L-T.1]) and, independently, in [L.1]. It is reproved in [L-L-T.1] using a multiplier technique based on the multiplier n(x) • V~p, where n(x) is a smooth vector field on 1) which extends the normal derivative v on F. Indeed, in the treatment of [L-L-T.1], the trace regularity result (2.19) for the homogeneous problem for (2.18) may be viewed as the starting step in building up the regularity theory of Theorem A for the nonhomogeneous problem (2.1).

We also note that for problem (2.18) the inequality 2

k~v/ d £ >

,

constrll{~ °, ~

(*)

T sufficiently large on the full £, or else on the portion £1 but only for the class of triplets {f~, F0, F1} satisfying the more restrictive radial vector field assumption (1.9), is explicitly given in [H.1] (private communication to the author by J. L. Lions) and it is also essentially contained in the estimates on the uniform stabilization result of Theorem B in the earlier work [L-T.3], albeit in a less transparent form, as the multiplier analysis of [L-T.3] is applied not to the homogeneous problem (2.18) but to the more complicated problem (see [L-T.3, equation (3.23)]) p , = Ap + F

pO, p ~ a t t = 0 p=0

in (0, oo) x f~, inll, in (0, ~ ) x F,

(**)

where F is a "bad" nonhomogeneous term related to the feedback (1.6). It was the presence of this term F that much complicated the multiplier analysis of [L-T.3] with multiplier e - ~ ' h ( x ) • Vp for (**) and forced assumption (a) below (1.8) on f~ that h(x) be parallel to v on F. Otherwise, with F = 0 , the radial vector field X-Xo would have sufficed to satisfy assumption (b) below (1.8) and the analysis given in [L-T.3] for problem (**) (equation (3.23) to inequality (3.59) in [L-T.3]) would reproduce inequality (*). See "Note added in proof" in [L-T.3, p. 387]. It is therefore recognized now that for the hyperbolic dynamics (1.1), the three problems of regularity theory, exact controllability, and uniform stabilization can all be handled under a common multiplier approach. Proof of Lemma 2.2. Step (i). With h ( x ) c C2(I)) the vector field assumed in Theorem 1.1, we multiply both sides of (2.18a) by h. V~, see Remark 1.4.

Exact Boundary Controllability on La(I~) x H-l(f~)

253

Proceeding as, say, in [L-L-T.1] and [L-T.3], we obtain the identity

Ov = fo HV¢. V¢d~+½fo(~2,-lv¢lm) divhdQ+[(~t,h"

V¢)a],=0,'=r (2.20)

where H(x) is the matrix in (1.12). (To make the present paper self-contained, we provide in Appendix A a derivation of (2.20).)

Step (ii).

Using the Dirichlet homogeneous B.C. (2.18c) we have: i

~,---0

on£;

V~±F;

h=(h" v)v+(h'r)r

onF,

IVy[ = [a~-v~ on £, hence,

(2.21a)

h. Wp = (h. v) Oq~,

Ov

(2.21b)

r = unit tangent vector, whereby the left-hand side (L.H.S.) of (2.20) then becomes L.H.S. of (2.20) =½

~vv h. v dE.

(2.22)

Step (iii). To estimate the second integral on the right-hand side of (2.20), we multiply both sides of (2.18a) by ~ div h and integrate by parts. Using (2.18c) and the identity V~p. V(~p div h) = ~pV(div h) • V~ + IVy[ 2 div h

(2.23)

we obtain f

(~,2-1V~12) div J (?

h dQ = [(~,,

~ div h)a]0r+ f . / (?

~V(div h)" V~# dQ.

(2.24)

For future reference we note that the above argument yielding identity (2.24) holds for an arbitrary div h (not necessarily the divergence of the postulated vector field h). Thus specializing to div h -= 1 (i.e., multiplying (2.18a) simply by ¢) gives the identity f

O

(~,~-IV~I =) dQ = [(~,, ~)~1~

(2.25)

to be used below. Thus, by (2.24) the right-hand side (R.H.S.) of identity (2.20) becomes

R.H.S.of(2.20)=f HV~o.V~odO+lf Q

~V(divh)'V~dO+JSo.r,

(2.26)

O

where/3o.r (boundary term at t = T and t = 0) is /3o.r = ½[(~o,, q~ div h ) ~ ] [ + [(~o,, h" V~)a]or.

(2.27)

254

R. Triggiani

Step (iv). Using Schwarz and Poincar6's inequalities on the right of (2.25) yields

I[(~,, ~o).]~1-< 4-G/2{I]~,(T)II~ + IIV~(T)II~+ I1~'1]~+ IIV~°II~}-<4-GE(0), (2.28) where as in (1.14) lifo112<- Cp 11]Vq~II1~;

E(t)=-ll~,ll~+lllV~olll~=-E(O).

(2.29)

Thus, by (2.28), we obtain from (2.25)

f

Q (,d-lV~,l =) dQ <<-~--~pE(O).

(2.30)

Similarly, using Poincar6's inequality on the first term on the right of (2.27) gives by (2.29) I[(¢,, ¢ air h)a]~'l- 2Dh4~pE(0),

(2.31)

2Dh ------m_ax]div hi;

(2.32)

Y~

and likewise by (2.29) 1[(¢,, h. V~o)n]ff[-< MhE(O),

(2.33)

Mh =--max]hi.

(2.34)

Thus, by virtue of (2.31) and (2.34), we estimate flo,r in (2.27) by

Iflo,TI-< [Dhx/~p+ Mh]E(O). Step (v). We have for the second

(2.35) term in (2.26) via Schwarz and Poincar6

inequalities

½It ~V(divh)'V~pd~>--2Ghfn'~P"V~'d">--2Ghx/-~pfn]V~'2d", (2.36)

4Gh =--m_axlV(div h)l. f~ Thus, using assumption (1.10) on (2.26), we obtain

(2.37)

H(x), and the estimates (2.35) and (2.36) in

R.H.S. of (2.20) -----(p-2Gh~-~p) f ]7q~]2 dQ do

KhE(O),

Kh ----Dh 4"-C--~p+ Mh.

(2.38) (2.39)

Step (vi). Using assumption (1.10) in (2.22), we obtain \~]

h. v dE, - L.H.S. of (2.20).

(2.40)

Exact Boundary Controllability on L2(I~)xH-l(l"l)

255

THUS, combining (2.38) and (2.40) yields

0

½ , \-~v/ h.

IV I2 dQ-KhE(O).

(2.41)

Recalling (2.30), we have

Io lV~ol2dO>- fo ~2t dO-V'--~pE(O).

(2.42)

Therefore, by (2.42)

folV12dQ>-½folVl=dO+½ = 2TE ( 0) -

d

2 E(0)

(2.43)

recalling the right identity in (2.29).

Step (vii).

Inserting (2.43) into (2.41) gives

\-~v h

VdZl>-[p-2Ghq~p](

E(O)-KhE(O),

T

(2.44)

hence \ "~u]

v d Y"' >- [ P - 2 G h~/-C~P]

(p- 2GhV~-~p)4~-~p+ 2Kh] J (2.45)

Thus, setting To as in (1.16), results in (2.17a) as desired, from which (2.17b) follows via (2.17c). []

3.

Illustrative Examples

The simplified test (sufficient condition) (1.19) of Theorem 1.1 (same as condition (1.9) of Theorem C) when specialized to radial vector fields readily allows us to construct many examples of applications. See, e.g., Example 3.3 given below. However, the main goal of the present section is to present classes of domains f~, where the full power of Theorem 1.1 with a general vector field yields a stronger and more precise conclusion than it is possible to obtain by applying the simplified test (1.19) with any radial vector field: the active portion F1 of the boundary which we shall obtain by virtue of Theorem 1.1 will be strictly smaller than the active portion of the boundary which is obtainable via the vector field test (1.19) (if we insist on including at least a subportion of such F~). Example 3.1. by

In

R 2 consider

h~(x)=x2+2ax+b,

the two-dimensional vector field h = [ hi, h2] defined

h2(y)=y2+2ay+b,

(3.1)

256

R. Triggiani

which we shall restrict to the first quadrant x, y -> 0. Here, a, b are two positive constants to be specified below. (i) We have div h = 2 x + 4 a +2y; V(div h) = [2, 2], IV(divh)l---x/8

Gh=X/8/4.

i n R 2, and

(3.2)

(ii) The matrix H and the constant p are

_ 0 I' H = 2x÷2a0 2v+2a

in x,y>-O,

p=2a

(3.3)

since Hr. v=2ix+a)v~+2(y+a)v~>-2a[v[ 2. Thus, in view of (3.2), for any bounded domain inx, y >-0 we select the positive constant a sufficiently large as to satisfy condition (1.13) of Theorem 1.1, with Cp a Poincar~ constant associated with such domain. (iii) Family of envelopes (curves) of vectorfield h. These are curves having at each point the vector h as tangent vector (not necessarily normalized). Thus, we must integrate dy/h2 = dx/h~. After making the assumption f12_ b - a 2 > 0, we obtain (or verify) that the family of envelopes is

y + a =fl(x+a)+fl2P fl - ( x + a)p '

O<-x< f l - a p , p

(3.4a)

with p parameter,/3 positive square root (see Figure 3.1). If we restrict to the region y - x - 0, the relationship between the parameter p and the point y ( 0 ) = y(x = 0) where the envelope meets the positive y-axis is p =/3y(O)/(b + ay(O)) so that in y-> x-> 0 the family of envelopes is given by

Y=

x(b + 2ay(0)) + by(O) b-xy(O) '

b y(O)'

0-< x < - -

(3.4b)

in terms of the parameter y(0). The slope along the envelope hitting the positive y-axis at y(0) is

dy bh2(y(O)) dx - (b - xy(O)) 2"

(3.5) [ i I

I

b/y(o)

×

Figure 3.1. (Example 3.1) Family of envelopes of h in (3.]).

Exact Boundary Controllability on L2(fl )

x H-I(I))

257

I I I

~A'

/ /

/

/

/

/

/

/

/

A /

// / / #'/ ¢ / // //

#

/

// t,

/

/

ParallelZ~// Figure 3.2. (Example 3.1) Definition of domain fL (iv) Definition of domain lq in y >-x >-O. With reference to Figure 3.2, let A: (XA = O, YA) be a point on the positive y-axis and let B: (xn, Ys = YA), XA < XB, be a point on the segment y =-YA. Consider the two envelopes qga and c~8, oriented along the vector field h, through A and B, respectively. By (3.1) the slope h2/hl decreases strictly, if x increases and y is held constant. Thus slope of tangent line= bh2(yA) > slope of tangent line to ~A at point A (b--Xya) 2 to ~B at point B. Since the slope along cOB increases strictly and continuously (cf. (3.5)), there is a unique point A' on qgn where slope of tangent line to cgB at point A' = slope of tangent line to CCAat point A. Let now C: (Xc, Yc) be a point on c~B strictly above (or on) the point A' and let D: (xo, Yo) be the point on CCAwith Yo = Yc. Then, the domain 1~ is defined as the region surrounded by 0~ = {arc AB} u {arc BC} u {arc CD} u {arc DA}, where the arcs A B and CD are chosen entirely within the region comprised between qga and qgB so that al~ is smooth. (v) Application of Theorem 1.1 to domain ~. By construction, with h the vector field in (3.1) and v the unit outward normal, we have h. ~, = 0

on arc A D and on arc BC,

h.~,<0

on arcAB,

h.~,>0

on a r c C D .

258

R. Triggiani

Thus, according to Theorem 1.1 (whose application is justified by the analysis in (i) and (ii) above), we conclude as follows: Claim 3.1(1).

In order to obtain exact controllability o n L2(I'~) × H - l ( f l ) , it is enough to exercise the control action u only on the (active) portion F1 of the boundary Ol'l given by F1 = arc CD ( whereby we can take Fo = {arc A D } u {arc AB} u {arc B C } in the notation of ( 1.1 )). (vi) Analysis of Claim 3.1(1) in light of test (1.19) for radial vector fields. The crux of the presently constructed class of domains 12 is the following: Claim 3.1(2). Claim 3.1(1) above cannot be obtained by restricting to radial vector fields and employing condition (1.19). More precisely, if we insist on exercising the control action u on at least a portion of the arc CD, then application of condition (1.19) yields the conclusion that, in fact, we should apply the control action u on a portion of Ol'l strictly greater than the arc CD, which contains the arc CD and, in addition, at least the arc CA'.

Proof of Claim 3.1(2). such that ( P - Po) " ~'p - 0

We must show that there cannot exist a point Po in on {arc A D } u {arc AB} u {arc BC}

R 2

(3.6)

if P c alq and ~,p denotes the corresponding outward unit normal at P. First, Po would have to lie below the line y =YA due to requirement (3.6) on the arc AB. Analysis on arc AD. With reference to Figure 3.3, we divide the halfplane Y<--YA in three closed regions IA, IIa, l l I a , by using the rays ra and ro

O

Figure 3.3. (Example 3.1) Analysis on arc AD for Po~ region I A

or

Po~ region IIIA.

Exact Boundary Controllabilityo n L2(~) x H-t(I))

259

defined as:

rA (ray tangent to CCAat A):

Y=

h2(YA) b X+yA,

x--<0, (see (3.5))

!

rD (ray tangent to ~A at D):

(

bh2(yA)

~Y = ( b ~ )

2

(x-xo)+yo,

!

[Y <-YA, IA-~((x,Y):~x+Ya<--Y<--YA}, IIA------- (x,y): (b_XDyA)2(X--Xo)+Yo <--y<--min YA,

(b - XoyA) 2 If P e arc A D we have vp : [ - h 2 , hl]/X/~2 + h 2.

(3.8)

Then: 1. If PoeregionIA~(P-Po).Vp<-O, with equality only if Po on rA and P = A. (3.9) . If Poe region I I I A ~ ( P - - P o ) " l'v-----0, with equality only if Po on rD and P = D. (3.10) . If Poe region IIA, then--with reference to Figure 3.4--let Q = Q(Po) be the unique point on ~A such that the line PoQ is tangent to ~¢A at Q (existence and uniqueness of Q is guaranteed by: slope of rA--
T a n g e n t at

Figure 3.4.

(Example 3.1) Analysis on arc AD for Poe region IIA.

260

R. Triggiani

Then, see Figure 3.4, the following two cases hold: (i) If P~ ~ arc A Q ~ ( P I

- Po) " vp, >_O, with equality only at P~ = Q.

(3.1 la) (ii) If P2 ~ arc Q D ~ ( P 2 -

Po) " z'P2<~ 0, with equality only at P2 = Q-

(3.11b) The above claims (3.9)-(3.11) follow easily from (cf. (3.8)) ( P - Po)" ~P = [xp - Xo, yp - y o ] " I - h 2 ,

hl]/4~+ h~

= { - h2(xp - Xo) + h l (yp - y o ) } / x / ~ .

(3.12) In fact, to see (3.9), replace Po with the point P~ situated where the vertical through Po meets the tangent line to P, see Figure 3.3. Thus, yp - Y o <- Yp - Y ~ and (3.12) gives

(P-Po)" ~',.<-(P-P~)" ~ , = 0 . To see (3.10), we replace likewise Po with the point Pg situated where the vertical through Po meets the tangent line to P, see Figure 3.3. Thus y , - Yo->Yp- Yg and (3.12) gives (e-Po)"

~'e>-(P-P'~) • r,=0.

Similar considerations prove (3.11a and b), see Figure 3.4. A n a l y s i s on arc B C . With reference to Figure 3.5, we define likewise regions I~, IIB, III~, by using the rays rs, rc : r8 (ray tangent to ~ at B), rc (ray tangent to ~B at C).

Tangent of

Figure 3.5. (Example 3.1) Analysis on arc

BC for Po e region II 8.

Exact Boundary Controllability on L2(~) × H-l(l))

261

Now, however, if P e arc BC, we have 2

2

~'p = [hE, -h~]/~/ h l + h 2

(3.13)

instead of (3.8). Using (P - Po)" ~'e = {h2(xp - Xo) - hl(yp - Y o ) } / ~

(3.14)

we likewise find for the arc BC: 4. If Po~ region I B m ( P - Po) • Vp>-O with equality only if Po on rB and P = B. (3.15) 5. If P o e r e g i o n I I I B ~ ( P - P o ) . vp<-O with equality only if Po (3.16) on rc and P = C. 6. If Poe region lie, then let Q = Q(Po) be the unique point on ~B such that the line PoQ is tangent to ~ga at Q. Then, using (3.14), we deduce likewise: (i) If P I e arc B Q ~ ( P ~ - P o ) " up <-0, with equality only if/'1 = Q. (3.17a) (ii) I f / ' 2 e arc Q C ~ ( P 2 - Po) • Vp >_O, with equality only if P2 = Q(3.17b) Conclusion of proof o f Claim 3.1(2). From the above analysis it follows that in order for a radial field P - Po to satisfy condition (3.6), we must have Po belong to the intersection {region

IA} N {region IIIB}.

(3.18)

- Bur this intersection is empty by construction, since the point C on cOB was chosen strictly above (or on) the point A'. Notice, in conclusion, that if we insist in exercising control action u at least on a portion of the arc CD, then the "best" result via the use of a radial vector field and of test (1.19) is obtained by letting Po lie on the ray rA and letting its coordinate Yo,~-oo. The inactive portion Fo of 0f~ (where we can take u --- 0) then contains {arc A D } u {arc A B } and, in addition, a larger and larger portion of the arc BA'. Control action u should then be exercised not only on arc C D but, in addition, at least on the arc A'C, This arc A ' C is superfluous using the analysis based on the vector field (3.1). []

Remark 3.1. The same argument shows that if, instead, the point C on cOB is chosen strictly below the point A', then the set in (3.18) is not empty. Thus, by choosing Po to be any point in the set described by (3.18), we obtain in this case of C strictly below A' on the curve cOB a vector field P - P o which yields the same conclusion as Claim 3.1(1). Example 3.2. defined by

In

hl=ax+by;

R 2 we

consider the two-dimensional vector field h = [hi, h2]

h2=cx+dy;

(3.19)

which again we shall restrict to the first quadrant x, y-> 0. Here, a, b, c, d are constants to be specified below.

262

R. Triggiani

(i) We have IV(div h ) l - 0,

hence

Gh = 0.

(3.20)

(ii) The matrices H and H + H* are H = Ia c

~l • "

H+H*=

12a b+,c

b+c 2d

and we then require H + H* to be positive definite a>0,

4ad-(b+c)2>O.

(3.21)

Thus assumptions (l.11) and (1.13) of Theorem 1.1 are guaranteed for any bounded domain with Poincar6 constant Cp. To simplify the computations we now specialize to a - d, b = c, with a > b > 0, i.e., to the vector field h~ = ax+ by;

h2 = bx+ ay.

(3.22)

(iii) Family of envelopes of the field h in (3.22). We integrate the (homogeneous) equation d y / d x = h2/hx and find that the family of envelopes is given by

[y--xl"+b=p(x+y) a-b,

y,x>--O,

p parameter. Restricting further to y - x-> 0 and setting y(0) = y ( x = 0) for the point of intersection of an envelope with the positive y-axis, we obtain that the family of envelopes is given implicitly by (y--x)a+b=(y(O))2b(x+y) ~-b, y>_x>_O, (3.23) in terms of y(0) as a parameter. To solve (3.23) explicitly we further specialize the situation by choosing a =3, b =½, (3.24) and setting y(0) = 2)7(0). We then obtain the family of envelopes y=(x+)7(O))+,/4x)7(O)+)72(O),

y>-x>-O.

(3.25)

The slope along an envelope in (3.25) is dy = 1 q2)7(0) dx x/4x)7(0)+)72(0) strictly decreasing from d y / d x [ x = o = a / b = 3 at x = 0 moreover, that for h as in (3.22)

h2 hi

•strictly decreasing on each line y = const ~ 0 froma/b=3atx=Otob/a=)asx'r+~ , strictly increasing on each line x = const ¢ 0 from b / a = ½at y = 0 to a / b = 3 as y 1' + ~ .

(3.26) to 1 as x~'+oo. Note,

(3.27) (3.28)

The family of envelopes in x, y - 0 is given in Figure 3.6. (iv) Definition of domain 1) in y >--x >-O. With reference to Figure 3.7 let A: (XA, YA), yA > XA > O, and let B: (xn, yB = ya), O< Xn < XA. Consider the envelope c¢a and ~B through A and B, respectively, oriented along the vector field h. Since, by (3.27), 1 < slope of tangent ray rA to CgAat point A < slope of tangent to c£n at point B < a / b

Exact Boundary Controllability on L2(f~) x H-~(D)

y~

I

0

~

263

///,////y~×

dd~Lxx=o = ~"

dY y=o ~ b--d'x 0 Figure 3.6.

X

(Example 3.2) Family of envelopes of h in (3.22) and (3.24).

and since, by (3.26), the slope along c¢B decreases continuously and strictly as x 1' +oo there is a unique point A' on c¢n where slope of tangent ray to c~B at point A' = slope of tangent to cCA at point A. Let now C: (Xc, Yc) be a point on qg~ strictly above (or on) the point A' and let D: (xo, yo) be the point on q¢A with YD = Yc. Then the domain f~ is defined as the region surrounded by

OD = {arc AB} u {arc BC} w {arc CD} w {arc DA}, where the arcs AB and CD are chosen entirely within the region comprised between ~ga and cOB so that aD is smooth. /

Yl

/

~ e ~

rc Tangent to ~o/ /lID_ ",, ,~// ~t //¢

at C

D"/" /

//

/

/// /// ///I I /// / / / I/ ' ~ . ~" rA Tangent to c~A Parallel 'b"~ at A rK Tangent to ~e ot A'

Figure 3.7.

(Example 3.2) Definition of domain D.

264

R. Triggiani

(v) Application of Theorem 1.1 to domain 12. By construction, with h the vector field in (3.22) subject to (3.24) and v the unit outward normal, we have h. i, = 0

on arc A D and on arc BC,

h.i,<0

onarcAB,

h.v>0

onarcCD,

and Theorem 1.1 then gives: Claim 3.2(1). In order to obtain exact controllability on L2(12)x H-~(12) it is enough to exercise the control action u only on the (active) portion F~ of the boundary 012 given by F~ = arc CD. (vi) Analysis of Claim 3.2(1) in light of test (1.19)for radial vector fields. The crux of the presently constructed class of domains 12 is the following claim, which is a verbatim restatement of the corresponding Claim 3.1(2) of Example 3.1, except that it now applies to Example 3.2. Claim 3.2(2). Claim 3.2(1) above cannot be obtained by restricting to radial vector fields and employing condition (1.19). More precisely, if we insist on exercising the control action u on at least a portion of the arc CD, then application of condition (1.19) yields the conclusion that, in fact, we should apply the control action u on a portion of 012 strictly greater than the arc CD, which contains the arc CD and, in addition, at least the arc CA'. The p r o o f of the above Claim 3.2(2) is similar to'the one o f Claim 3.2(1) in Example 3.1 and is therefore omitted. With reference to Figure 3.8, we note only that the candidate point Po of the sought radial vector field P - P0 must belong to the set R c c~ R A which, by construction, is empty since C was chosen on ~B strictly above (or on) the point A'. The "best" result via the use of a radial vector

PA" ~t. i I Parallel ~y

Figure 3.8. (Example 3.2) Regions RA and Rc.

Exact Boundary Controllability on L2([), ) × H-1(II)

265

A

o

B

Figure 3.9. Inactive part F o where u = 0 may be any arc strictly smaller than one-half the circumference.

field and of test (1.19) is obtained (if we insist on exercising control action u at least on a portion of the arc CD) by letting Po lie on the ray rA and letting its coordinate Yo~ -oo. This way we obtain an inactive part Fo of the boundary which contains {arc AD} u {arc AB} and a larger and larger part of the arc BA'. Example 3.3. The radial vector field test (1.9) = (1.19) is particularly simple in the case of l-I being a unit disk: then, as the active portion F1 of the boundary we can take any continuous arc of the circumference, which is strictly greater than one-half a circumference. See Figure 3.9. Can F1 be taken to be exactly one-half a circumference? Suppose a general (smooth) vector field h(x, y) in R 2 is tangent to the unit circumference at two diametrically opposite points, say x = 0 , y = 1, and x = 0 , y = - l . Then we have h2(0, 1 ) = h 2 ( 0 , - 1 ) = 0 , and thus there exists some ~7, - 1 < r / < l , where h2y(O, r/) =0. Thus, the matrix H + H * fails to be uniformly positive definite on the disk, yielding an inconclusive case, as far as Theorem 1.1 is concerned. 4.

Variable Coefficient Case

The content of the present section, in particular of Remark 4.3 below, benefits from correspondence that I. Lasiecka and the author have had with J. L. Lions while working in the spring of 1986 on the exact controllability problem for the wave equation with control action in the Neumann boundary conditions. This work will be reported elsewhere [L-T.5]. There are interesting new phenomena, and related new difficulties, that arise when we consider the exact controllability problem for the generalized wave equation with (smooth) variable coefficients. See Remark 1.3. To illustrate this p o i n t - - a n d indicate a general strategy for solution--it suffices to analyze the simplest case. Consider

' yw, = Aw w ( 0 , . ) = w °,

in Q,

(4.1a)

w , ( 0 , . ) = w I inl~,

(4.1b)

wl~0= 0 Wlz , = U E L2(~1)

in Eo,

(4.1c)

in 21,

(4.1d)

instead of (1.1), where y = 3,(x) = smooth - 3'0 > 0.

(4.2)

266

R. Triggiani

The (maximal) regularity theory for (4.1) in L2(fl) x H-1(II) is likewise covered by [L-L-T.1], [L.3], and [L-T.2]. Then, as we have seen in Lemma 2.1, exact controllability on L2(~) x H-l(l~) of (4.1) in [0, T] is equivalent to proving the inequality (same as (2.10))

fE, \(~0@~2 / dZL-C~II{~°,

@,..21 J't[no(a)xL2(a)

(4.3)

for some C~-> O, where now tp solves the homogeneous problem

{

y~o,t = aq~,

~o(0, .) = ~°~ H~(f~),

(4.4a) q~,(0, .) = tpx c L2(I~),

¢pl~-= 0,

(4.4b) (4.4c)

which replaces (2.18). The generalization of Lemma 2.2 and Theorem 1.1 is now:

Theorem 4.1. Given the triplet {1~, Fo, F1} assume that there exists a vector field h(x) = [hi(x) . . . . , h,(x)] ~ C2(1)) such that the following three conditions hold: (i)

h. v

<-0 on Fo,

1, = outward unit normal.

(ii) f M ( x ) v ( x ) . v(x)dll>_p f [v(x)12, dO, 3n Ja

(4.5) for some constant p > O, forall v(x) ~ L2(O; R"), (4.6)

where the matrix M ( x ) is given by M ( x ) = H(x)-t h ( x ) . Vy(x) I, 2y(x)

(4.7)

with H given by (1.12). ( A sufficient, checkable condition for (4.6) to hold is that M ( x ) + M* ( x ) be uniformly positive definite on ~. ) (iii) p > 2Gh,~X/-~p (this may be removed, see Remark 4.3), (4.8) where 4 G h , , = n ~ x lV(div h)+ v(h'--~Vy y) l ,

(4.9)

Cp = Poincare constant in (1.14). Then there is To> 0 (to be specified below) such that if T > To, then inequality (4.3) holds true for problem (4.4) with C'T = ClCh,p,y, where cl is as in (2.17c) and C h.a.z,= p -- EGh,rX/~p .

(4.10)

ThUS, for T > To, problem (4.1) is exactly controllable on L2(lq) x H-l(f~) as in the conclusion of Theorem 1.1.

Exact Boundary Controllability on L2(~ ) x H - t ( l l )

267

For To we can take

To -

2Kh,~,

(4.11)

p _ 2 Gh,vV/-~p ~-v/-~P '

gh.,y = Dh.~,X~p + Mh,v "

'

_mrl ,

x/~

'

(4.12) (4.13)

Mh., = max 4~lhl. f~

Remark 4.1. If we specialize h(x) to a radial vector field h(x) = x - x o , Xo~ R ", then H(x)=--identity and the condition

p=min

( x - Xo)" v ~,(x) +1>0 27(x)

(4.14)

is sufficient for (4.6) to hold with p defined by (4.14). In this case we have, see (1.18), div h --- n and from (4.9) and (4.12)

4Gh,~ = n~x Iv ([ X- Xo] . V y) ] ; 3'

(4.15)

2Dhv' =m~x v/yn÷ [X--Xo]" VV ~

"

Because of (4.2), we see that condition (4.8) holds in particular for IV3'1 sufficiently small on l).

Proof (Sketch). 1. Multiplying (4.4a) by h. V~ (see Remark 1.4) and integrating by parts as in Appendix A we obtain -~

h. vd~,=

Q

H V ~ . V~p d Q + [(7¢,, h. Vtp)t~]or

+½IQ~2div3,hdQ-1IQlV~[2divhdQ

(4.16)

after using the B.C. (4.4~). This is the counterpart of (2.20)-(2.21). 2. In the third integral on the right of (4.16) we use div yh = h. VT+ y div h, thus obtaining

R'H'S" °f (4"16) = Io HV ~ " V ~ dQ + ½IQ ~2 h " V T dO + ½f Q 7~2' div h dO Iv~l 2 div h dQ+[(7~o,, h. V~o)a]or.

-~ Q

(4.17)

268

R. Triggiani

3. Multiplying (4.4a) by ~ div h and integrating by parts as in step (iii) of Lemma 2.2 yields

fo '/~Pt2div hdQ- fo [V~I 2 div h dQ =f

O

~oV(divh).V~pdQ+f

~

0~o

'/divh[~o~,]ffdfl-f /divhd2~ J~0v/ (4.18)

after using the B.C. (4.4c). 4. Inserting (4.18) into (4.17) results in R.H.S. of(4.16)= I

Q

HV~o.V~odO+½f q~:h.VydO Q

+½ f eV(div h) • Vq~dQ JQ +½[(q~,, ? div hq~)u]~'+ [(yq~t, h. Vq~)a]or,

(4.19)

which reveals a new term--the second integral on the right of (4.19)--over (2.26)-(2.27). To handle this new term, we proceed as follows. 5. (New step over the proof of Lemma 2.2.) Multiplying (4.4a) by (h. Vy/'/)tp and integrating by parts we obtain

Ic ~o2h.V'/dO=fo ,Vq~,2h'VYdO+Iq ~oV~o'v(h'-~-VyY) dO + [(~t, (h"

fVOv' //"d Z31j

(4.20)

after using the B.C. (4.4c). 6. Inserting (4.20) in (4.19) and using assumption (4.5) on Fo gives, finally, ½ , \0--~] dI~I~>L.H.S-of(4.16)=R.H.S-of(4.16)

=fQ[H+ h"2,/V'/ i]Vq~. V¢dO +½foq~V~o.[V(divh)+v(h'-h-VyY)]dO+ao.r, 1 ao, r = ~[(¢t, ['/div h + h. V'/]~o)a]or + [('/q~t, h" Vq~)u]or,

(4.21) (4.22)

whose right-hand side is the counterpart of (2.26)-(2.27). 7. From here on the proof proceeds as in Lemma 2.2, steps (iv)-(viii) using now the "energy"

E(t) =-fJu '/~o,2+IV~o[2 dl'l~ E(0)

(4.23)

Exact B o u n d a r y Controllability on L2(I~) x H-~(I~)

269

(verified, as usual, by multiplying (4.4) by ¢, and integrating by parts). Details are omitted. [] Remark 4.2. A study of condition (4.6) for the matrix M(x) in terms of a suitable vector field h(x) has not been performed yet. We may perhaps seek h(x) in the form h(x) = Vf for a suitable f. Remark 4.3. During the recent direct study (i.e., without passing through stabilization) of the exact controllability problem for the wave equation with control action in the Neumann boundary conditions [L.6], [L-T.5], some strategies have emerged that have a bearing also on the Dirichlet problem (1.1) (or on problem (4.1), or even on a fully general hyperbolic problem, for that matter). First, it turns out that an alternative approach is available in handling the second integral at the right of (2.26), which manages to eliminates altogether the "smallness" condition (1.13) (resp. (4.8)) of the vector field h, and to replace it with the uniqueness condition of the homogeneous problem (say for problem (4.1)): ( ~/tp, = A~ ] ~ --- 0

on (0, T] x l~ = Q, on (0, T] x F,

(4.24)

~t

T sufficiently large, say T > Tu ~ q~~ 0

in Q.

What we lose in this approach, however, is that the estimate of the time To> 0 such that for T > To exact controllability is achieved is not as precise as under assumption (1.13) (resp. (4.8)). Second, the uniqueness property (4.24) is always true if Fo = O by virtue of the Holmgren-John theorem. If, instead, Fo# 0 , we understand that ongoing work by Littman [L.9] should provide the required nontrivial extension of the Holmgren-John uniqueness theorem from the constant coefficient case [H.2, Theorem 5.3.3, p. 129] to the variable coefficient case. 5.

The Case with Neumann Homogeneous B.C. on ]~o

In this section we consider the question of exact controllability for the problem

{

w, = Aw

w ( 0 , . ) = w °,

in Q,

(5.1a)

w , ( 0 ; . ) = w ~ inQ,

(5.1b)

I

0w[ -=0 Ov I~o w]~, = u ~ L2(0, T; L:(F~))

inEo,

(5.1c)

in El,

(5.1d)

where Fo # O and F1 # Q, which was raised by R. Glowinski at the Workshop in Montreal, 6-9 October 1986. The following counterpart of Theorem 1.1 h o l d s true for problem (5.1).

270

R. Triggiani

Theorem 5.1. Suppose that {ft, Fo, F~} possesses a vector fieM [ h i ( x ) , . . . , h,(x)] ~ C2(l)) such that the following three conditions hold:

h(x)=

(i) h. u = 0 on Fo, r,=(outward) normal; (5.2) 6 (ii) same as assumption (1.11)-(1.12) in Theorem 1.1; (5.3) (iii) same as assumption (1.13)-(1.14) in Theorem 1.1 (see Remark 4.3). (5.4)

Then there is To>O---given by (1.16) of Theorem 1.1 in fact--such that if T> To then, for each pair {w°,wl}~L2(fl)XHr~,(l)), there exists some u~ L2(0, T; L2(F1)) -= L2(E1) such that the corresponding solution of (5.1) satisfies w( T) = w,( T) = O and

Iw(t)

w,(t) e C

(

L2(I~) ) [0, T]; Hr~(ll) "

Here we have H~,(II) - { f ~ H l ( f l ) : ) q r , = 0};

Hr~(II) = [H~,(fl)]'.

(5.5)

Proof of Theorem 5.1.

We shall indicate the modifications that need to be made on the arguments of Section 2 in order to obtain Theorem 5.1.

Step O. The solution at time T to problem (5.1) with homogeneous initial data w ° = w I = 0 is given by

w( T; t = O, w ° = O, w' = O) IA ~ S( T - t)D,u( t) dt wt(T;t=O, wO=O, wl=O ) =.LPTU=,A~f c ( T _ t ) D l u ( t ) d t '

(5.6)

counterpart of (2.1). Here, D~ is the operator defined by the mixed problem

fay=0

inl),

D l g = y ¢=~ JAY=0 y=g

inF0,

(5.7)

onF1,

counterpart of (2.2), for whose well posedness we refer to the vast literature, quoted, e.g., in [M.1, pp. 233-235]. Moreover, A is now the positive, self-adjoint operator defined by A f = - A f, where

~(a)={fcH2(l)):~

a/-' I Fo

=f,r, =0}.

(5.8)

In fact, by Green's theorems, A is initially symmetric, with dense domain 29Cooo(lq), and positive since F1 ~ Q. Moreover,

/Ifll ~¢A"2~=-IIm'/=fll = =

(Af, f )

= f [Vfl 2 dl), dot

Vf~ ~ ( a 1/2) -- HL,(I~)

6Condition (5.2) is much stronger than the corresponding condition (1.10) of Theorem 1.1. However, (5.2) seems indispensable at the level of analyzing (5.18) below. Note that the conditions on h imply r 0c~F1 # ~, see Remark 1.4.

Exact BoundaryControllabilityon L2(fl) x H-~(ll)

271

(first for f ~ ~(A), and then extended), and the generalized Poincar6 inequality holds from f = A-~/ZA~/2f so that

Ilfll=~ c~ ff~ Ivfl = da,

G = IIA-'nll •

Thus, A is also semibounded below

(Af f) >--~2p(f f),

V f c ~(A).

Hence, the von Neumann-Friedrichs theorem applies [D-S, 1, Vol. II, p. 1240] and there is a self-adjoint extension with the same lower bound, still denoted by A. The self-adjoint calculus then implies that - A generates a strongly continuous cosine operator C(t) and S(t)=~'oC(r)dr, t~R, on L2(ll) (see [F.1]). (For simplicity, we keep for A, C ( . ) , and S ( . ) the same symbols as in Section 2.) Now let D* be the adjoint operator of D~: (Dig, x)L~(a)= (g, D*x)L2WO. The counterpart of Lemma 2.0 is then: Lemma 5.2.

Forf ~ ~)(A) we have

_ D , A f ={2f/Ov

on

F1,

on Fo.

(5.9)

Proof. With g ~ L2(F0 we proceed as below (2.3); we use Green's theorem and apply (5.7)-(5.8) (subscripts refer to L2-norms): -(D*Af g)r, = (Af D~g)n

\

Ov / r ,

Step 1. Exact controllability of problem (5.1) on the space L2(f~) x HF~(fl) over [0, T] is equivalent to the ontoness property dgT:

L2(X~)

onto, L2(FL)x H r ' ( I ) ) ,

in turn equivalent to the property that the Hilbert space adjoint ~ * has a continuous inverse

I~T zlZI0 L2(.~,I)~CTII{ZO,Zl}IIL2(n)×H~:(O.), for a positive Cr independent on [Zo, Zx].

(5.10)

272

R. Triggiani

Step 2. The counterpart of Lemma 2.1 is now: Lemma 5.3.

(a) For [Zo, zl] c L2(II) x Hrl(II) we have

=D*a[c(t-r)a-'z,+S(t-r)(-Zo)] on~,.

(5.11)

Here 4` solves the backward problem

4`,, = A4`,

(5.12a)

4`1,=~ = 4`0, 4`,[,=~ = 4`,,

(5.12b)

0_0 so = 0,

(5.12c)

4`1~1 =0,

(5.12d)

4`°=A-'z,~H~(lI)=~(A1/2);

4`' = -Zo~ L2(l)),

(5.13)

counterpart of (2.7)-(2.9) and (2.13). It is given explicitly by

4`(t; 4,°, 4`') = C ( t - r)4`°+S(t - T)4`'~ C([0, T]; H~,(II)= ~(A'/2)),

(5.12e) G(t; 4`04`,)= C ( t - T)4` 1- A S ( t -

T)4`°~ C([0, T]; L2(I))),

(5.12f)

where the indicated regularity is the result of standard cosine operator theory [F.1]. (b) For any 0 < T
f~ (04`) ~ 2 d~l ->C~'ll{o°' 0 }']nll(a)xL2(a), 2 '

1

(5.14)

counterpart of (2.10).

Proof. Same as proof of Lemma 2.1 except that now we use (recall (5.5))

1

n r , ( l ) ) = ~(A'/2);

nr,'(l~) = [~(A'/2)] ',

(y, z)n~'m)= (A-ly, z) L:(m, counterpart of (2.11).

(5.15) []

Step 3. The counterpart of Lemma 2.2 is now: Lemma 5.4.

Under the same assumptions as in Theorem 5.1, there is To > 0, same as in the conclusion of Theorem 5.1, such that if T > To then inequality (5.14) holds true with C'T= Ch,p( T - To), Ch,p given by (2.17b).

Exact Boundary Controllability on L2(lq)x H-I(Iq)

273

Proof of Lemma 5.4. It is the same as the p r o o f of L e m m a 2.2 except for the following modifications. As there, let ~0 in (5.14) be, by time reversal, the solution to problem (5.12a, c, d) where now (5.12b) is replaced by ~(0)---~0°e H~. (f~), ~Ot(0) = qJle L2(fl). Returning to the fundamental identity (2.20), with q~ there now replaced by qJ, we have that its left-hand side is now L'H'S" °f ( 2"20) = Iz~ O--~( h " V t~) d~ + ½Iz~ ~O~h" u d~" - ½Iz~ ]v ~ol2h " u (5.16) We split ~ into Eo and ~;1 and use in (5.16) the following relations: (4/,---0, /

on E,:

qJ-= 0 ~

/IV~01

=~

,

(5.17)

We then obtain the counterpart of (2.22) (which shows the major novelty o f problem (5.1) over problem (1.1)):

(O~OI2h" ud~,l+½ f~ qJ~h" udEo

L.H.S. of(2.20)=½ I s ,

1

\Ou/

[

o

[vqd2h ~,d~o

(5.18)

dZ o

(it is here that the strong assumption (5.2) is invoked)

L (0j) -~

h. ~' dE1.

(5.19)

1

The right-hand side of (2.20), with q~ there now replaced by qJ, can be treated as before under the assumptions (5.3) and (5.4). On the other hand, (5.19) combined with the right-hand side of (2.20) and the a priori regularity (4.12e and f) gives plainly

,

c 11(6 °,

(5.20)

for any 0 < T < o o . Then (5.20) by duality as in [L-T.2] and [ L - L - T . 1 ] yields [w(t), w,(t)] ~ C([0, T]; L z ( f l ) x H r ~ ( f l ) ) . This way, T h e o r e m 5.1 is proved. []

Acknowledgment We wish to express our thanks to F. L. Ho for his careful reading, in the fall of 1986, of the first version of this paper; in particular, for suggesting the use of the Poincar6 inequality first and then the Schwarz inequality (instead of using the Schwarz inequality first and then the Poincar6 inequality as was done in the first version) in estimating the term/30,r in the proof of Lemma 2.2. This resulted in a slight improvement of the time TOin (1.16).

274

R. Triggiani

Appendix A. Sketch of the Proof of (2.20)

The identity div(~bh) = h. V~b+ ~bdiv h and the divergence theorem give I h.V~bdf~=Ir

~bh. vdF-I~

~bdivhdfL

(A.1)

Multiply both sides of (2.18a) by h. V~o and integrate in Q. As to the left-hand side, we integrate by parts in t, use ~o,h.V~, =½h. V(~o~), and identity (A.1) with ~b= 2 . We obtain

fQ ~,,h "V~ dO= I [~th "V~]ro df~-½ I ~2h " v d~ +½fo ~2 div h dO. (A.2) As to the right-hand side, we use Green's first theorem, the identity Vw. V(h. Vw)=

HVw. Vw+½h. V(IVw[2),

and identity (A.1) with ~b= IV~o[2. We obtain f e A~o(h. V~o) dO = f.l~Ov O~°(h" V~°) d ~ - IQ HV~o. V~odO

-½LlV~12h'vd~+½IQ[V~pl2divhdQ.

(A.3)

Equating the left-hand side (A.2) with the right-hand side (A.3) results in (2.20).

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.

Abstract Treatment. ~=My+~u,

Consider the abstract equation y(O) =yo,

(B.O)

M being the generator of s.c. semigroup of operators on the Hilbert space Y and ~ : U ~ ~ ( ~ ) ~ Y being a linear, generally unbounded operator from another Hilbert space U to Y, with M-1~ continuous from U to Y (without loss of generality we may assume that M is boundedly invertible). The solution to (B.O) with Yo= 0 is

~ru = M

e:a(r-t)M-l~3u(t) dt, (B.1)

~ru =

Io e~(r-')~u( t) dt

(conventional notation).

Exact Boundary Controllability o n L2(I'~) × H-I(I'I)

275

Let z be a target state in Y and consider the following minimization problem: Minimize

J(u)=½11u tlL2(o,T;u) 2 over all u ~ L2(0 , T, U) such that ~TU =Z, under the (exact controllability) assumption that there exists at least one such u. If we indicate by ( ( , ) ) the duality pairing between Y' and Y, the Lagrangian can be written as

L(u,p)=½(U,U)L2(O,T;U)--((p,~TU--Z)),

p c r'.

Taking Lu = 0 yields u ° = 5g*p °

(B.2)

z = &PTU° = ..~T~*p °,

(B.3)

thus

where we easily have ~ , p O = ~3,e~*(T-,)pO.

(B.4)

Specialization to the Hyperbolic Problem (1.1) with Fo = •.

In this case we have

[L-T.1 ]-[L-T.4]: Y = L2(n) x H-~(I~),

U = L~(F),

and with pO = [pO, pO] ~ y,: AO,pO = ~ . e~*(T-t) p O= ~3. ea(,- r)p O= ~)*[ ea(t- T)p O]2' (where /9 is defined in (2.2) and [Y]2 denotes the second component y2 of Y = {el, Y2})

= -i)as(t-

T)p°+ l ~ a C ( t - T)A-lp°;

hence by (2.3)

u°(t) = ~ ( t , o , 1 ) ,

(B.5)

or, where

{

~Pt, = A~

~l,=r=~°=--a-~p°eH~(12),

in Q, ~,l,=r=~l=p°eL2(ll)inf,,

~1~- 0

(B.6)

in E.

Moreover, from (B.3) we obtain > C ~llp 0II2~ ((pO, z)) = ((pO, .L~,r~.rpO)) = IILf*TP 0II2L2(~)--

(B.7)

for T > To > 0 by L e m m a 2.2. Define then z = Ap °

(B.8)

so that A is an isomorphism Y' onto Y by the Lax-Milgram lemma, and pO = [pO; pO] = A - ' z e Y'.

(B.9)

276

R. Triggiani

Thus, given the target state z, equation (B.9) provides the initial condition for problem (B.6) whose solution furnishes the optional steering control by (B.5). This is precisely the control used by Lions in his {F, F', A} method, where now F = H ~ ( n ) x L2(fl).

References [D-S.II [F.I] [G.il [H.1]

[H.2] [L.I] [L.2] [L.3] [L.4] [L.5] [L.6]

[L.7]

[L.83

[L.9] [L-L-T.1] [L-T.1] [L-T.2] [L-T.3]

[L-T.4]

[L-r.5] [M.I]

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Exact Boundary Controllability on L2(D ) × H-l(ft) [R.1]

[R.2] [R.3]

[T-L.1] [T.I]

277

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Accepted 20 July 1987