EXACT CONTROLLABILITY OF A BEAM IN AN INCOMPRESSIBLE INVISCID FLUID

It is well known that an Euler Bernoulli beam may be exactly controlled with a single control acting on an end of the beam In this article we show that for certain boundary conditions the same result holds for a beam that is surrounded by an incompressible inviscid uid with a su ciently small density The proof involves reducing the control problem to a moment problem and using compactness properties of the Neumann to Dirichlet map for the Laplacian operator to obtain the needed estimates Mathematics Subject Classi cation C K B


Introduction. It is well-known that an Euler-Bernoulli beam may be exactly
controlled by a single control acting at one end of the beam. However, unless the beam is in a vacuum, there is usually some amount of in uence that the surrounding medium will impose upon the beam. Thus it is natural to ask whether the controllability properties of a beam (or other elastic material) are preserved when its motion is coupled with that of a surrounding uid. Banks,et. al. 2] proposed a model describing the coupling between the acoustic pressure in a rectangular cavity and the vibrations of an Euler-Bernoulli beam positioned along one edge of the rectangle. In that problem, the motivation was to utilize controllers placed along the beam to stabilize the acoustic pressure inside the cavity. Although much progress has been made on this problem (see 3] for well-posedness, and references therein for stabilization of nite dimensional approximations also see Micu and Zuazua 8,9] for non-uniform stabilization results) it is impossible to exactly control the state of the system by controllers that are active only along one edge of the rectangle. This is due to the presence of \trapped rays" (or echos) (see Bardos,Lebeau,Rauch,1]).
In this article we i n vestigate the controllability results one can obtain for systems similar to the one in 2], however, for the case of an incompressible uid. In the incompressible case the uid motion is described by a potential equation, and hence there are no obstructions to controllability associated with \trapped rays". We s h o w that at least for certain boundary conditions, the exact controllability p r o p e r t i e s o f an Euler-Bernoulli beam (and other elastic models) are retained when the beam is surrounded by an incompressible uid of su ciently small density.
1.1. Problem Statement. At equilibrium we assume that a still, incompressible, inviscid uid occupies the region of the x{y plane with Lipschitz boundary ;. We assume that ; consists of a in exible part ; 1 and and a exible part ; 0 . I f ; 0 is surrounded on both sides by the uid, then ; 0 = ; + 0 ; ; 0 where ; + 0 and ; ; 0 represent the upper and lower surfaces of the beam, respectively. F or de niteness we suppose that ; 0 = f(x h=2) : 0 < x < 1g for some h > 0. Some possible domains are illustrated in Figures 1 and 2. The transverse displacement w(x t) (the displacement i n t h e y direction) satis es the Euler-Bernoulli beam equation: m @ 2 w @t 2 + EI @ 4 w @x 4 = F on (0 1) R + (1.1) where m denotes the density of the beam, EIdenotes the sti ness of the beam and F denotes the force exerted by the uid on the beam, all in per length units. The force F in (1.1) is the di erence in the uid pressures on the top and bottom of the beam, i.e., F(x t) = ;p(x h=2 t ) + p(x ;h=2 t ) on (0 1) R + : (1.2) The uid we consider is modeled from the theory of ideal uids, see e.g., 6]. The linearization (about a still state) of the Euler system describing the motion of an ideal uid is @q @t + rp = 0 div q = 0 where q = q(x y t) is the velocity eld, p = p(x y t) is the pressure, and (assumed to be constant) denotes the density per area of the uid. (We regard the uid as a two dimensional medium.) Taking the divergence of the rst equation gives p = 0 in R + (1.3) and matching the normal component of the uid acceleration to the acceleration of the boundary leads to the following boundary conditions (n is the outward unit normal to ;). @p @n = 0 on ; 1 R + (1.4) @p @y (x ;h=2 t ) = @p @y (x h=2 t ) = ; @ 2 w @t 2 (x t) on (0 1) R + : (1.5) In the above formulation, as in 2], the domain of the uid is approximated by the xed domain to avoid nonlinearities. Remark 1.

Main
Results. Let V = ff 2 H 2 (0 1) : f 0 (0) = f 0 ( 1 ) = 0 g H = L 2 (0 1): We will mainly be concerned with the following set of boundary conditions for the beam. @w @x (0 t ) = @w @x (1 t ) = @ 3 w @x 3 (1 t ) = 0 EI @ 3 w @x 3 (0 t ) = f(t) t > 0: (1.6) Initial conditions for the beam are of the form fw @w=@tg t=0 = fw 0 w 1 g 2 V H : The boundary conditions in (1.6) can be viewed as \sliding clamps" at each end. The clamp on the right end moves freely in the transverse direction (we are considering small motions) while a transverse force f(t) is applied at the left end.
Concerning the control problem (1.1), (1.6), (1.7), without the uid coupling (i.e., with F 0), it is well known that given any initial state fw 0 w 1 g 2 V H , and any f 2 L 2 (0 1), there is a unique solution w which satis es w 2 C( 0 1) V) \ C 1 ( 0 1) H): (1.8) Furthermore the system is exactly controllable on V H in any time T > 0 that is, given any initial state fw 0 w 1 g 2 V H , a n y terminal state fv 0 v 1 g 2 V H and any T > 0 there exists f 2 L 2 (0 T ) for which fw(T) @w @t (T )g = fv 0 v 1 g: The main result of this article are the following. Our approach to the proof of Theorem 1.1 is to reduce the control problem to a moment problem. (See 12, 13] for some early applications of this method). A necessary condition for exact controllability is that the eigenvalues of the system possess a uniform asymptotic separation. In the case = 0 the eigenvalues are known exactly. F or > 0 w e are able to show that the amount that the eigenvalues are perturbed by the uid is proportional to and to a certain norm of the \Neumann to Dirichlet map" for the Laplacian (see Section 2). Thus the problem of proving eigenvalue separation reduces to proving certain compactness properties of the Neumann to Dirichlet map. In Section 2 we s h o w that this map is continuous from L 2 to H 1 and this is precisely what is required in the case of boundary conditions (1.6) to prove eigenvalue separation.
The methods used in this paper are valid for certain other types of boundary conditions and also in situations where other dynamics are present on the exible boundary besides the Euler-Bernoulli beam. In particular, exact controllability results also hold when the Euler-Bernoulli beam is replaced by either a string equation or a Rayleigh beam. These extensions (and some limitations of our approach) are discussed in Section 5.
Remark 2.1. For a given g 2 L 2 (0 1), the solution p to (2.9) is only determined up to a constant, but 0 g is uniquely determined. Thus the condition R p d = 0 used in de ning (and hence also in de ning 0 ) in (2.1) provides uniqueness for p, but the forces acting on the beam are uniquely determined, independent of the number that R p d is set equal to. where = m and the subscripted variables represent di erentiation with respect to that variable. (Henceforth we adopt this notation.) We rst consider the case of homogeneous boundary conditions: De ne the sesquilinear forms In the variational formulation of (2.12), (2.13), we seek a function w 2 C( 0 1) V)\ C 1 ( 0 1) H) satisfying The dynamics on Z are those that describe rigid motions. Suppose fw 0 v 0 g 2 Z. Then B ;1 Afw 0 v 0 g = fv 0 0g. It follows that e B ;1 At fw 0 v 0 g = fw 0 + tv 0 v 0 g: (2.19) We summarize these facts in the following proposition. Proposition 2.3. B ;1 A is the in nitesimal generator of a strongly continuous group T = ( T t ) t2R on V H= E Z. The restriction of T to the invariant subspace E is unitary and the restriction of T to the invariant, two dimensional subspace Z is given by (2.19). Consequently, given the initial conditions fw w t g t=0 = fw 0 w 1 g in V H there exists a unique solution w to (2.12), (2.13) with w 2 C( 0 1) V) \ C 1 ( 0 1) H): Furthermore, for each t > 0 a(w(t) w (t)) + b(w t (t) w t (t)) = a(w(0) w (0)) + b(w t (0) w t (0)): The semigroup T in Proposition 2.3 may a l s o b e e x t e n d e d b y duality t o a n isomorphic semigroup on the space H V 0 . T o see this, we s e t X = X 0 = V H and X 1 = D(A) V . Then X 0 1 = H V 0 . Also, we h a ve that X 1 = E 1 Z, where E 1 = ( D(A) \Ṽ) Ṽ . T h us X 0 1 = E 0 1 Z, where E 0 1 is dual to E 1 with respect to the inner product e (and pivot space E). Thus we k eep the same de nition of T on Z. Since iB ;1 A is self-adjoint and continuous from E 1 onto E, it has a self-adjoint extension that is an isomorphism from E to E 0 1 . Therefore we de ne T on E 0 1 by duality, as follows: e(Ty Y) def = e(T(iB ;1 A) ;1 y (iB ;1 A)Y ) 8 y 2 E 0 1 8 Y 2 E 1 : Thus we h a ve the following.

Main estimates.
In this section we prove the main estimates relating to the eigenvalues and eigenfunctions of B ;1 A that will be needed for the proof of Theorem 1.1.
Recall that ( k ) 1 k=0 denotes the sequence of eigenvalues of B ;1 A, arranged in increasing order and ( k ) denotes the corresponding sequence of eigenfunctions, normalized so that b( k k ) = 1 , f o r a l l k. We let ( 0 k ) denote the eigenvalues of A, also arranged in increasing order and (e k ) denote the eigenfunctions of A, normalized so that (e k e k ) = 1 , a l l k. Explicitly, Proof. Since 1 = b( k k ) = k k k 2 + ( 0 k k ) a n d c 0 = 1 k 0 k L(H H 1 ) we h a ve 1=(1 + c 0 ) k k k 2 1 k = 0 1 : : : : (3.4) Recall that for all k, A k = k B k = k (I + 0 ) k and Ae k = 0 k e k = C 0 k 4 e k . Therefore for any k j = 0 1 : : : To estimate P k;1 j=0 j( k e j )j we consider two cases: k > l and k l. If k > l then we can write P k;1 j=0 j( k e j )j in the form k;1 X In the case k l we obtain k;1 X j=0 j( k e j )j = j( k e 0 )j + l;1 X j=1 j( k e j )j c 0 + p l ; 1: Inequality (3.3) follows from the above estimates. De ne the sequence (b k ) 1 k=0 by b k = h 0 k i = k (0) k = 0 1 : : :: (3.6) In the above, 0 denotes the Dirac delta function with mass at x = 0. Note that the sequence (b k ) i s w ell-de ned since 0 2 V 0 . Lemma 3.4. Let (b k ) be the sequence de ned in (3.6). Then: (1) (b k ) 2 l 1 , i.e., (b k ) is a bounded s e quence.
Before we p r o ve Theorem 1.1 we need to brie y discuss Riesz bases. A sequence (p k ) 1 k=1 is a Riesz basis for the Hilbert space X if it is the image under a Hilbert space isomorphism F : X ! X of an orthonormal basis. Corresponding to the Riesz basis (p k ) is a uniquely de ned biorthogonal sequence (q k ) 2 X which satis es (p k q j ) X = kj = 1 if k = j 0 if k 6 = j : The sequence (q k ) is itself a Riesz basis for X. If f 2 X then there is a unique sequence (f k ) 2 l 2 such that f = P 1 k=1 f k p k and there exists positive constants c, C such that ck(f k )k l 2 k fk X Ck(f k )k l 2 : In addition, f k = ( f q k ) X 8k 2 N: We refer the reader to Young 16] for details. Proof of Theorem 1.1. The corresponding rst order form of (4.4){(4.6) analogous to (2. The standard semigroup theory gives W 2 C( 0 T ] H V 0 ), for any T > 0.
However this is suboptimal and our rst goal is to show that (4.4) holds, i.e., W 2 C( 0 T ] V H ). Since W 0 2 V H , b y Proposition 2.3 we know that T t W 0 2 C( 0 1) V H ).
Thus, for purposes of proving that (4.4) holds, we m a y assume without loss of generality that W 0 = 0 : (4.10) Recall that B ;1 A has eigenvalues ( k ) for k = 0 1 2 : : :and corresponding eigenfunctions ( k ) which are normalized so that b( k k ) = 1, for all k. for appropriate constants K 1 : : : K l and again obtain the regularity in (4.4).
Finally, using that (4.8) holds on H V 0 and (4.9) holds on V H it is easy to verify that (4.5) and (4.6) are satis ed. The uniqueness follows from the uniqueness of the homogeneous problem. Now w e consider the control problem. Suppose we wish to nd a control which drives the initial state W 0 2 V H to the terminal state W T 2 V H in time T. Since (4.9) is valid on V H , the problem is the same as nding a control which drives the initial state 0 to the terminal state W T ; T T W 0 . Therefore it is enough to assume that W 0 = 0 n d f 2 L 2 (0 T ) for which W(T) = W T . Corresponding to W T is a uniquely de ned sequence ( k ) 2 l 2 with k( k )k l 2 CkW T k V H , for some C > 0. We wish to ndf 2 L 2 (0 T ) that solves the moment problem (4.14) (with t = T). By Lemma 3.4 there exists 0 > 0 such that if < m 0 then (jb k j ;1 ) 1 k=0 2 l 1 . T h us, for such , w e h a ve ( k =b k ) 2 l 2 .
By Lemma 3.2 the eigenvalue separation condition holds. Thus (p k ) forms a Riesz basis for M, its closed span in L 2 (0 T ). Let (q k ) denote the biorthogonal sequence to (p k ) i n M. W e letf = 1 X k=0 ( + k =b k )q + k + 1 X k=0 ( ; k =b k )q ; k : (4.18) It follows from (4.7) that the series forf converges in L 2 (0 T ) and it is easy to see from the biorthogonality thatf indeed solves the moment problem (4.14). This concludes the proof of Theorem 1.1.

Extensions and related results.
The methods used in this paper can be applied to several similar situations involving other types of boundary conditions and di erent dynamics on the exible boundary. W e conclude this paper with a brief discussion of the results one can obtain in some of these situations. Concerning the exact controllability however, our approach i s v alid only for boundary conditions. In Lemma 3.1 we needed to use the estimate u = m X j=k u j cos j x) ( 0 u u) L 2 (0 1) c k kuk 2 L 2 (0 1) where c is independent o f k and m. In other words, the Neumann to Dirichlet map, which w e know i s c o n tinuous from L 2 (0 1) to H 1 (0 1), is shown in (5.1) to also produce exactly one degree of smoothing in terms of Fourier series. However, in the case of simply supported boundary conditions: w(0 t ) = w(1 t ) = w xx (0 t ) = 0 w xx (1 t ) = f(t) then we w ould have t o p r o ve (5.1) with sines in place of the cosines. Unfortunately, unless we know that 0 maps into H 1 0 we can not conclude that an estimate like (5.1) holds. Therefore our approach fails to prove c o n trollability for types of boundary conditions involving xed e n d c onditions.
On the other hand, we can consider control of boundary conditions such a s : w xx (0 t ) = w xx (1 t ) = w xxx (0 t ) = 0 w xxx (1 t ) = f(t): (5.2) In this case the eigenfunctions of A satisfy all the properties that were required of the cosines, namely orthogonality, the same growth rate of eigenvalues and the property (5.1). Consequently a result like Theorem 1.1 is valid when the boundary conditions (4.2) are replaced by those in (5.2). (There is one di erence: with (5.2) we get exact controllability in a quotient space, where the the zero element of this quotient space is a one-dimensional space of rigid rotations.) We can also consider problems where the control acts on the moment instead of the shear for example, in (4.2) set w xx (1 t ) = f(t) a n d w xxx (1 t ) = 0. Again we nd that for su ciently small , the exact controllability is the same as when = 0. The proof involves only some minor modi cations.

String equation on exible boundary. This problem was considered in
the compressible case by Micu and Zuazua 8,9]. In the incompressible case the problem can be written w tt + 0 w tt ; w xx = 0 on (0 1) R + (5.3) w x (1 t ) = 0 w x (0 t ) = f(t) on R + (5.4) w(x 0) = w 0 (x) w t (x 0) = w 1 (x) on (0 1) (5.5) where fw 0 w 1 g are given in V H = H 1 (0 1) L 2 (0 1). (We adapt the previous notation to this problem, rather than make new notation.) For convenience, we have set all the physical parameters in (5.3){(5.5) to 1, other than , the uid density. All the steps in the analysis of this problem are the same as in the analysis with the Euler-Bernoulli beam, with the exception that the eigenvalue estimates are weaker, due to the fact that when = 0 the spectrum is uniformly spaced. We were able to prove the following result. Again, we encounter di culties if instead of (5.4), xed end-conditions are imposed.
In (5.7) we consider the case of xed end conditions with moment control, although our method applies equally well to other boundary conditions.
In the analysis of this problem it is unnecessary to utilize the compactness of 0 due to presence of the term w xxtt it is enough to utilize only the continuity o f 0 . F or this reason, boundary conditions involving xed ends do not present a n y di culties. By following the same steps as in the proof of Theorem 1.1 (and slight modi cations for the simply supported boundary conditions) we w ere able to prove the following result.