Spectrum and Exact Controllability of a Hybrid System of Elasticity.

Transcription

1 Spectrum and Exact Controllability of a Hybrid System of Elasticity. D. Mercier, January 16, 28 Abstract We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. The spectrum of the spatial operator involved in this evolution problem is studied. The associated eigenvectors are also characterized. Using the HUM method and the expression of the solution in terms of Fourier series, we prove that the hybrid system is exactly controllable in an arbitrarily short time by means of only one controller in the usual energy space. Key words. Hybrid system, Exact Controllability, Spectrum, Fourier series, Flexible beams. AMS 37N35, 35P15, 35P2, 35Q72, 93C2. 1 Introduction In this work we consider a hybrid system consisting of an elastic beam of length L, clamped at one end and attached at the other end to a rigid antenna, whereon is applied a dynamical control v 1. In the case of ususal initial data the exact controllability is proved in [7],[8] for a beam of limited length L. The aim is to extend this result for any length L thanks to the characterization of the associated spectrum. The vibration u(x, t) of the beam is governed by the Euler-Bernoulli equation, and the oscillations u(l, t), u x (L, t) of the antenna are described by the Newton-Euler equations through which the control dynamics is filtered: u tt + u xxxx = < x < L, t > u(, t) = u x (, t) = t > ρu tt (L, t) u xxx (L, t) = t > Ju xtt (L, t) + u xx (L, t) = v 1 (t) t > u(x, ) = u (x), u t (x, ) = u 1 (x) < x < L Laboratoire de Mathématiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, VALENCIENNES Cedex 9, FRANCE, s : denis.mercier@univ-valenciennes.fr 1 (1)

2 where ρ > is the mass and J > is the moment of inertia of the antenna. In the case of usual initial data : u H 2 (, L), u 1 L 2 (, L) the regularity of weak solution is insufficient to define the traces u tt (L, t) and u xtt (L, t). Following the method developped in [8] we transform the system (1) into an abstract system { utt + Au = v u() = u, u t () = u 1 where the state u is defined by u(x, t) = (u(x, t), u(l, t), u x (L, t)) and the control v by v = (,, v 1 ). A is a positive definite operator in the product space H = L 2 (, L) IR 2. Using the classical Hilbert Uniqueness Method (HUM), the control problem is reduced to the obtention of an observability inequality. In [8], a multiplier technique is employed to obtain this inequality. That was done in the case of two controllers (i.e an additional control v in the third equation of (1)). In the same way, with only one control as in (1), the same technique leads to the desired inequality but for a beam of limited length. To our knowledge this problem is not solved yet for a beam of unspecified length. In this work we solve this problem where the method consists in developing the solution of the associated homogeneous problem in terms of Fourier series. Let us quote [1] where the authors used this technique successfully for a problem of two Euler-Bernoulli beams connected by a point mass (see also [6] for the study of the controllability of a network of beams with interior point mass). The paper is organized as follows: Section 2 is devoted to the setting of the problem. Section 3 is devoted to the characterization of the eigenvalues and eigenvectors of our problem. In Section 4, inequalities of observability are established for the solution of the homogenous problem with usual intial data and smooth initial data. In Section 5 we state the results of controllability by resuming work of [8]. 2 Data and Framework We study the following problem control: u tt + u xxxx = < x < L, t > u(, t) = u x (, t) = t > u tt (L, t) u xxx (L, t) = t > u xtt (L, t) + u xx (L, t) = v 1 (t) t > u(x, ) = u (x), u t (x, ) = u 1 (x) < x < L, (2) where we assume, without loss of generality, that the positive physical constants ρ and J are equal to 1. Following [7], we first write formally the system (2). We start by writing (2) in the form: (u(x, t), u(l, t), u x (L, t)) tt + (u xxxx (x, t), u xxx (x, t), u xx (L, t)) = (,, v 1 (t)) (u(x, ), u(l, ), u x (L, )) = (u (x), u(l, ), u x (L, )) (u(x, ), u(l, ), u x (L, )) t = (u 1 (x), u t (L, ), u xt (L, )) According to this formulation we define the product space H = L 2 (, L) IR 2 endowed with usual inner product (.,. ) H by (U 1, U 2 ) H = L u 1 (x)u 2 (x)dx + ξ 1 ξ 2 + η 1 η 2, U i = (u i, ξ i, η i ) H, i = 1, 2, 2

3 and define the operator A on the Hilbert space H by { U = (u, ξ, η) : u H D(A) = 4 (, L), ξ = u(l), η = u x (L) u() =, u x () = }, (3) AU = (u xxxx, u xxx (L), u xx (L)), U = (u, ξ, η) D(A). (4) The following result concerning the operator A is proved in [8]. Lemma 1 (Properties of the operator A) The operator A defined by (3)-(4) is a nonnegative self-adjoint operator with a compact resolvant. 3 Spectrum Our aim is to characterize the spectrum σ(a) of A. In particular, we are interested in the asymptotic behaviour of the eigenvalues and the eigenvectors of A. According to Lemma 1 this spectrum is positive and discrete. Let us start by describing the equation checked by the elements of σ(a) (namely the characteristic equation) and also the associated eigenvectors. 3.1 Characterization of the eigenelements Proposition 2 (Characteristic equation). Let 2 σ(a), >, be an eigenvalue of A then satisfies the characteristic equation where the 2 2 matrix M(, L) is defined by f( ) = detm(, L) =, (5) M(, L) = ( a 11 a 12 a 21 a 22 ), (6) with a 11 = s 1 ( ) + e L s 2 ( ) + e 2 L s 3 ( ) a 12 = s 1 ( ) + e L s 4 ( ) e 2 L s 3 ( ) a 21 = s 5 ( ) + e L s 6 ( ) + e 2 L s 7 ( ) a 22 = s 5 ( ) + e L s 8 ( ) e 2 L s 7 ( ) (7) 3

4 and s 1 () = s 2 () = cos( L) sin( L) s 3 () = s 4 () = cos( L) sin( L) s 5 () = s 6 () = cos( L) + sin( L) s 7 () = s 8 () = cos( L) sin( L) Moreover, the multiplicity of 2 is 1 and the associated eigenvector U = (u, u (L), u x (L)) has the following expansion: u i=4 (x) = c i e i (x), x [, L], (9) where and i=1 e 1(x) = cos( x) e 2 (x) = sin( x) e 3 (x) = e L e x e 4 (x) = e x c 1 = a 12 c 2 = a 11 c 3 = 1 2 e L (c 1 + c 2 ), c 4 = c 1 e L c 3 Proof. Let 2 σ(a), ( > ) be an eigenvalue of A with associated eigenvector U = (u, ξ, η ) D(A). From the definition (3)-(4) of A that means that u satisfies the conditions (12) to (16) hereafter : u xxxx (x) = 2 u (x), < x < L (12) (8) (1) (11) u xxx (L) = 2 u (L) (13) u xx(l) = 2 u x(l) (14) u () = (15) u x() = (16) 4

5 As the fundamental solutions e i, i = 1,..., 4 of the fourth order derivative (12) are given by (1), then u may be written as i=4 u(x) = c i e i (x), x [, L]. i=1 for some unknowns c i, i = 1,.., 4. Since the transmission and boundary conditions (13)-(16) are equivalent to a system of 4 homogeneous linear equations, we get a 4 4 homogeneous system of equations. Owing to the fact that the conditions (15)-(16) are equivalent to { c4 = c 1 e L c 3 c 3 = 1 2 e L (c 1 + c 2 ), (17) and imposing the transmission conditions (15)-(16), this reduces our system to a 2 2 homogenous one. We may write this last system as follows: M(, L) ( c1 c 2 ) =, (18) with M(, L) given in (6) and where (7)-(8) are obtained after some computations. (18) has non-trivial solution if and only if (5) holds. Now, we observe that e a 11 = [cosh( L) + L sinh( L) ] [cos( L) + L sin( L) ]. L L Since: u >, cosh(u) > 1, sinh(u) > 1 and sin(u) < 1, the previous identity shows that u u a 11 >, >. Consequently rank(m( (, L)) ) and we deduce that the multiplicity of a 2 σ(a) is 1. If 2 σ(a), the vector 12 is clearly solution of (18) thus we get (11). a Asymptotic behaviour of the spectrum As announced before, our goal is to use the development in Fourier series of the solution to get controllability results. To this end, the asymptotic behaviour of the characteristic equation (5) as + is of great interest. For our next purposes let us denote by { 2 k} k IN the monotone increasing sequence of eigenvalues of A and for all k IN, let U k = (u k, u k(l), u k x ) be the associated eigenvector given by (9) in Proposition 2. Proposition 3 (Asymptotic behaviour of the characteristic equation) The characteristic equation (5) given in Proposition 2 is equivalent with cos( L) + cos( L) + sin( L) + sin( L) cos( L) + g( ) =, (19) 5

6 where g( 1 ) = o( ) (i.e lim + g( ) = )). Thus, the asymptotic behaviour of the spectrum σ(a) corresponds to the roots of the asymptotic characteristic equation cos( L) =. (2) In other words, there exists an integer k with k IN, k = 1 L (π 2 + k π + kπ + ǫ k ) (21) with lim k + ǫ k = Proof. From Proposition 2, f( L) = a 11a 22 a 12a 21. Then we remark that in (7) the functions s i ( ), i = 1,..8 are bounded on [1, + ). Thus after some computations we see that the function f has the following expansion f( ) = e L [s 1 ( )s 8 ( ) + s 2 ( )s 5 ( ) s 4 ( )s 5 ( ) s 1 ( )s 6 ( ) + o(e L )] (22) But s 1 ( )s 8 ( ) + s 2 ( )s 5 ( ) s 4 ( )s 5 ( ) s 1 ( )s 6 ( ) = cos( L) + cos( L) + sin( L) + o( 1 ) Inserting the previous indentity in (22) and multiplying the characteristic equation by e L we obtain (19). Finally we can prove that (19) implies (21) using the same argument employed in Example 7.2 of [2]. Example 4 The asymptotic behaviour of the eigenvalues is illustrated in figure 1. where we have plotted the function f( ) = e + L f( ) in a interval of length 9π 2 as well as the points ( π + kπ) in the same interval in the case L = Asymptotic behaviour of the eigenvectors To obtain the observability inequality, we also need to study the asymptotic behaviour of the eigenvectors. More precisely, let 2 σ(a) and U = (u, u (L), u x (L)) be the associated eigenvector. We have to study the asymptotic behaviour of u x (L) 2 with respect to U 2 H. The main result is summarized in the following proposition: Proposition 5 Let 2 k σ(a) and U k = (u k, u k(l), u k x (L)) be the associated eigenvector given in Proposition 2, then lim k + U k 2 H = L 4 and there exist constants α 1 > and α 2 > such that the following estimate holds (23) α 1 2 k (u k x (L)) 2 α 2 2 k (24) 6

7 Figure 1: The characteristic equation with L = 1 Proof. In this proof, for the sake of simplicity, we remove the index k in the writing of k. Let us introduce some useful notations: for each σ(a), C = (c 1, c 2, c 3, c 4) represents the vector of the coordinates of u in the fundamental basis (e i ) i=1,...4 given in Proposition 2. Let us also define the three 4 4 matrices : L M1 = ( e i (x)e j (x)dx) 1 i,j 4 M2 = (e i (L)e j (L)) 1 i,j 4 M3 = ((e i ) x(l)(e j ) x(l)) 1 i,j 4. Then we have and U 2 H = C (M 1 + M 2 + M 3 )(C ) T, (u x (L))2 = C M 3 (C ) T. From Proposition 2 and using a formal calculation software we get: C = 1 2 (1, 1, cos( L) + sin( L), 1) (1, 1, cos( L) sin( L), 1) + O(e L ) In the same way, using the fundamental basis (1) we may write: L M1 = 1 L 2 + O( 1 ), 7 (25)

8 M2 = M3 = cos( L) 2 cos( L)sin( L) cos( L) cos( L)sin( L) sin( L) 2 sin( L) cos( L) sin( L) 1 sin( L) 2 cos( L)sin( L) sin( L) cos( L)sin( L) cos( L) 2 cos( L) sin( L) cos( L) 1 + L O(e ), + L O(e ), where in the previous identities O( 1 ) (resp. O(e L )) represents a matrix with all its terms of order Now, from (19), 1 (resp. O(e L )). Thus, after computations we arrive at U 2 H = L 4 + ((1 + )cos( L) + sin( L)) 2 + O( 1 ) (26) (u x(l)) 2 = ((1 + )cos( L) + sin( L)) 2 + O( 1 ) (27) ((1 + )cos( L) + sin( L)) 2 = 1 2(cos( L) sin( L) g( )) 2 (28) and using at the same time (2) and the fact that g( ) = o(1), we arrive at the conclusion. Remark 6 If we normalize the eigenfunctions so that U k 2 H = 1 then, by Proposition 5, the estimate (24) is still true. 4 Observability In this section we prove some observability results which are consequence of the asymptotic properties of the previous section. The reason to study these properties is that, by means of the so called HUM method (see [4],[5]), controllability properties can be reduced to suitable observability inequalities for the adjoint system. Since in the sequel the solution will be expressed in terms of Fourier series, the observability inequality will be proved using Ingham inequality (cf. [3] or [1]): Theorem 7 (Observability inequality and spectral gap) Let k be a sequence of real numbers such that there exist (α, β, k ) IR 2 IN satisfying k+1 k α >, k k (29) and k+1 k β >. Consider also T > π/α. Then there exist two constants C 1 (T) and C 2 (T) which only depend on α, β and k such that, if f(t) = k Z α k e i kt, it holds for all (α k ) l 2 (IR). C 1 (T) α k 2 k Z T T f(t) 2 dt C 2 (T) α k 2 k Z 8

9 As (2) is a self-adjoint system we are reduced to the same system, without control. Therefore, consider system (2) without control: U tt + AU = U() = U U t () = U 1 (3) Since A is self-adjoint, positive definite and since A 1 is compact, using the spectral decomposition theory, we can define the powers A 2α L(D(A α ); D(A α )) for any α IR, where the domain D(A α ) is defined by: D(A α ) = {U : U = u k U k with u k 2 ( k ) 4α = U D(A α ) < }, k IN k IN where the sequence (U k )k IN represents an orthonormal basis in H with respect to the spectrum σ(a). In particular we have V = D(A 1 2 ) = { U = (u, ξ, η) : u H 2 (, L), ξ = u(l), η = u x (L) u() =, u x () = U 2 V = L U 2 xx }, (31) dx, U = (u, ξ, η) V. (32) Proposition 8 Let (U, U 1 ) H V and consider U = (u, ξ, η) the solution of (3) with initial data (U, U 1 ). Then for each T > there exist constants D 1, D 2 > which only depend on T such that D 1 ( U 2 H + U 1 2 V ) T u 2 xt (L, t)2 dt D 2 ( U 2 H + U 1 2 V ) (33) Proof. Let us first write the inequalities (33) in terms of Fourier series: The spectral Theorem allows to write U(t) = sin( k t) (u k cos( k t) + u 1k )U k (34) k IN k where the u ik s are defined by U i = k IN u ik U k, i =, 1. The identity (34) may be written in the following form U(t) = 1 c k e i kt U k (35) 2 k Z provided that we set c k = u k i u 1k k if k > c k = c k if k < k = k if k < U k = U k if k < 9

10 Therefore T u 2 xt (L, t)2 dt = 1 4 T c k k u k x (L)ei kt 2 dt k Z Now let us remark that from Proposition 2, we know that all eigenvalues of A are of order 1 and from (21), lim k + k+1 k = +. Taking this remark into account we can apply Theorem 7 with any α > ; thus we deduce for all T > : C 1 c k 2 2 k u k x (L)2 k Z From (24) in Proposition 5, we get α 1 C 1 k Z c k 2 T T Since H = D(A ) and V = D(A 1 2) we have k Z c k 2 = 2 k IN (u 2 k + u2 u 2 xt (L, t)2 dt C 2 c k 2 2 k u k x (L)2 k Z u 2 xt (L, t)2 dt α 2 C 2 1k 2 k which leads directly to the conclusion, with D i = α i C i, i = 1, 2. k Z c k 2 (36) ) = 2( U 2 H + U 1 2 V ), (37) Proposition 9 Let (U, U 1 ) V D(A) and consider U = (u, ξ, η) the solution of (3) with initial data (U, U 1 ). Then for each T > there exist constants D 1, D 2 > which only depend on T such that D 1 ( U 2 V + U 1 2 D(A) ) T u 2 x(l, t) 2 dt D 2 ( U 2 V + U 1 2 D(A) ) (38) Proof. The proof is similar to the one of Proposition 8, where we get instead of (36). Finally, since c k C 2 1 k Z 2 k T u 2 x (L, t)2 dt C 2 c k 2 k Z 2 k (39) we arrive at (38). c k 2 k Z 2 k ( u2 k 2 k IN k = 2 + u2 1k ) = 2( U 4 2 V + U 1 2 D(A) ), (4) k 5 Exact Controllability We complete this work by stating two controllabillity results with initial data in different spaces. With usual initial data we have: Theorem 1 Let T >. Then for all (U, U 1 ) V H, there exists a controller v 1 (H 1 (, T)) such that the weak solution U(t) = (u(t), u(l, t), u x (L, t)) of the controlled problem (2) satisfies the final conditions u(t) = u t (T) =. (41) 1

11 Proof. The proof is exactly the one we find in [8] since it is based on the observability inequality (33) Now, we consider the control with smooth initial data: Theorem 11 Let T >. Then for all (U, U 1 ) D(A) V, there exists a controller v 1 L 2 (, T) such that the weak solution U(t) = (u(t), u(l, t), u x (L, t)) of the controlled problem (2) satisfies the final conditions (41). Proof. As previously we use the proof in [8] with the help of the observability inequality (38) Acknowledgments The author would like to thank S. Nicaise for his remarks and the rereading of this work. 11

12 References [1] C. Castro, E. Zuazua. Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass. Mathematical and Computer Modelling, 32, p , 2. [2] B. Dekoninck, S. Nicaise. Control of networks of Euler-Bernoulli beams. ESAIM : Control, Optimisation and Calculus of Variations, 4, p , [3] A. Haraux. Séries lacunaires et contrôle semi-interne des vibrations d une plaque rectangulaire. J. Maths Pures et Appl., 68, p , [4] J.L. Lions. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Masson, [5] J.L. Lions, E. Magenes. Problèmes aux limites non homogènes et applications I-III. Dunod, Paris, [6] D. Mercier, V. Régnier. Control of a network of Euler-Bernoulli beams. J. Math. Anal. and Appl.,27, to appear. [7] B. Rao. Contrôlabilité exacte frontière d un système hybride en élasticité. C.R Acad. Sci. Paris, 324, Série 1,1997, [8] B.Rao Exact Boundary Controllability Elasticity of a Hybrid System by the HUM method. ESAIM, COCV, 21, Vol.6, p