Arnaud Münch

Farid Ammar-Khodja

Giuseppe Geymonat

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–634http://france.elsevier.com/direct/CRASS1/Partial Differential Equations/Calculus of VariationsOn the exact controllability of a system of mixed orderwith essential spectrumFarid Ammar-Khodja a, Giuseppe Geymonat b, Arnaud Münch aa Laboratoire de mathématiques, Université de Franche-Comté, UMR CNRS 6623, 25030 Besançon, Franceb Laboratoire de mécanique et de génie civil, Université de Montpellier II, UMR CNRS 5508, 34095 Montpellier, FranceReceived 2 April 2008; accepted 11 April 2008Available online 23 May 2008Presented by Évariste Sanchez-PalenciaAbstractWe consider the exact boundary controllability problem for the system y′′ + Acy = 0 where Ac is a self-adjoint operator in Hwhose (non empty) essential spectrum is {0}, which implies the non-exact controllability. We construct the HUM control for allinitial condition in the subspace H⊥c × Ṽ⊥0 of H × V′. Each subspace H⊥c and Ṽ⊥0 is spanned by the eigenvectors associated tothe eigenvalues of Ac satisfying the separation condition (Σ), and this property seems general. We also study the behavior of thecontrol when c goes to zero. To cite this article: F. Ammar-Khodja et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméSur la contrôlabilité exacte d’un système d’ordre mixte ayant un spectre essentiel. On s’intéresse à la contrôlabilité exactefrontière d’un système du type y′′ + Acy = 0 où Ac est un opérateur dans l’espace H dont le spectre essentiel (non vide) est {0}, cequi implique la non contrôlabilité exacte. On construit le contrôle HUM pour toute donnée initiale dans le sous-espace H⊥c × Ṽ⊥0de H × V′. Les sous-espaces H⊥c et Ṽ⊥0 sont engendrés par les vecteurs propres associés aux valeurs propres de Ac satisfaisant lacondition de séparation (Σ), propriété qui semble générale. Nous étudions également le comportement du contrôle lorsque c tendvers zéro. Pour citer cet article : F. Ammar-Khodja et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans [4] on a étudié la contrôlabilité exacte de (4) et on a montré l’existence des données initiales (u0,u1) qui nesont pas exactement contrôlables. Dans le but de caractériser le sous-espace des données initiales qui sont exacte-ment contrôlables, on s’intéresse dans cette Note à un exemple particulier 1D. Soient c ∈ R+ et ω = (0,1) ; on veutdéterminer les données initiales (y0,y1) ∈ H × V′ où H = L2(ω) × L2(ω) et V = H 10 (ω) × L2(ω) telles que pourtout T > 0 suffisamment grand il existe un contrôle v dans L2(0, T ) pour le système :E-mail addresses: farid.ammar-khodja@univ-fcomte.fr (F. Ammar-Khodja), geymonat@lmgc.univ-montp2.fr (G. Geymonat),arnaud.munch@univ-fcomte.fr (A. Münch).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.04.009630 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–634(Sc)⎧⎨⎩y′′ + Acy = 0 dans ω × (0, T ),y1(0, t) = 0, y1(1, t) = v(t) dans (0, T ),y(ξ,0) = y0, y′(ξ,0) = y1 dans ω,(1)i.e. des conditions initiales telles que l’unique solution de ce problème vérifie,y(ξ, T ) = 0,y′(ξ, T ) = 0 dans ω.L’opérateur auto-adjoint Ac est associé à la forme bilinéaire ac définie sur V × V par,ac(y,v) =∫ω(y1,1(ξ) + cy2(ξ))(v1,1(ξ) + cv2(ξ))dξ, (2)où yi,1(ξ) = ∂yi(ξ)/∂ξ . L’étude est ramenée à l’analyse de la propriété d’observabilité pour le problème homogèneadjoint :⎧⎨⎩φ ∈ C(0, T ;V) ∩ C1(0, T ;H),φ′′ + Acφ = 0 in (0, T ),(φ(0),φ′(0)) = (φ0,φ1)(3)à données initiales dans V × H . L’opérateur Ac a pour spectre {0, c2, c2 + k2π2}, k > 0. Il apparaît que la valeurpropre 0 est de multiplicité infinie de sorte que 0 appartient au spectre essentiel σess(Ac) de l’opérateur, tandis queles valeurs c2 et c2 + k2π2 sont de multiplicité 1. On peut représenter la solution de (3) par le développement ensérie (8). L’utilisation d’un résultat d’Ingham permet alors d’obtenir l’observabilité de (3) pour toute donnée initialedans Ṽ0,⊥ × H̃⊥0 , où H̃⊥0 , définie par (11), est l’orthogonal du noyau de Ac et Ṽ0,⊥ est l’orthogonal de ce noyau dans V(Proposition 2.1). Il est à remarquer que pour ces données intiales la solution φ de (3) vérifie φ(t) ∈ Ṽ0,⊥ pour toutt > 0. Par ailleurs, la présence de la valeur propre c2 empêche l’observabilité uniforme lorsque c tend vers zéro illus-trant le caractère singulier du mode v0 (Proposition 2.2). En utilisant la méthode HUM, [7], on obtient la contrôlabilitéde (1) pour toute donnée initiale (y0,y1) ∈ H⊥c × Ṽ⊥0 (voir Théorème 3.1) où Ṽ⊥0 est la fermeture de Ṽ0,⊥ dans V ′.On remarquera que H⊥c , resp.Ṽ⊥0 est le sous-espace de H, resp.V′, engendré par les vecteurs propres associés auxvaleurs propres de Ac satisfaisants la condition de séparation (Σ ), voir (19). On sait, [4], que la calotte membranairesphérique n’est pas, en général, exactement contrôlable. Dans ce cas aussi, avec une condition de séparation du mêmetype (Σ ), on peut identifier les données initiales exactement contrôlables, [1].1. Statement of the problemIn [4] it has been considered the question of the exact controllability for a system of the form:⎧⎨⎩u′′ + Au = 0, in ω × (0, T ),u(0) = u0, u′(0) = u1,Bu = v on ∂ω × (0, T ),(4)when A is a Douglis–Nirenberg elliptic system of mixed order, B is a system of normal boundary conditions; therealization of A associated to the boundary conditions B in the Hilbert space H is defined by a sesquilinear selfadjointform a(u,v) on V × V. Exact (or null) controllability means that at a time T large enough the solution verifies:u(T ) = 0, u′(T ) = 0. (5)It has been proved that when the operator A has a block of zero order, and hence AB has an essential spectrum(see [5]), there exist some initial data (u0,u1) such that the system (4) is not exactly controllable. This is the situation,in particular, of membrane shells of arbitrary shapes. In this context we are interested in the following question:(EC) Characterize the space of initial data (u0,u1) such that the system (4) is exactly controllableAs a preliminary step in this Note we consider a particular 1D example. Let be c ∈ R+, ω = (0,1) and y(ξ) =(y1(ξ), y2(ξ)) for ξ ∈ ω. Let us consider in H = L2(ω) × L2(ω) the operator Ac associated to the bilinear formac(y,v) defined on V × V by (2) (where V is a suitable subspace of H). One can take for simplicity V = H 1(ω) ×0F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–634 631L2(ω). For c = 0, the bilinear form is simply a0(y,v) =∫ωy1,1v1,1 dξ . In this Note we study the null-controllabilityof (1). Let us remark that the operator Ac is of Douglis–Nirenberg type with a zero order block and hence the resultsof [4] apply. We characterize the class of initial data such that the system (Sc) is exactly controllable; we also discussthe behavior of the control as the parameter c goes to zero.2. Spectral property and observability2.1. Spectral property and decompositionA simple computation gives: σ(Ac) = {0, λ0 = c2, λk = c2 + k2π2, k = 1,2, . . .}. SinceKer Ac ={vζ = (ζ,−c−1ζ,1) ∈ H , ζ ∈ H 10 (ω)}, (6)and the eigenfunctions associated with λ0 = c2 and λk = c2 + k2π2 are respectively:v0 = (0,1), vk =√2kπ√λk(sin(kπξ),ckπcos(kπξ)), (7)one deduces that σess(Ac) = {0}. Let us remark that an orthonormal (in H) basis of Ker Ac is:wk =√2c√λk(sin(kπξ),−kπccos(kπξ)), k = 1,2, . . .and that {wk,v0,vk} is an orthonormal basis in H.The weak solution of the problem (3) can be expanded as follows, setting μ0 = c and μk = √λk :φ(t) =∞∑k=1(ak + bkt)wk +(A0 cos(μ0t) + B0 sin(μ0t))v0 +∞∑k=1(Ak cos(μkt) + Bk sin(μkt))vk, (8)where the coefficients ak, bk,A0,B0,Ak,Bk are determined from the expansion of the initial data:φ0 =∞∑k=1akwk + A0v0 +∞∑k=1Akvk, and φ1 =∞∑k=1bkwk + μ0B0v0 +∞∑k=1μkBkvk. (9)The assumption (φ0,φ1) ∈ V × H implies that∞∑k=1(a2k + A2k)< ∞,∞∑k=1(cak + kπAk)2 < ∞,∞∑k=1(b2k + λkB2k)< ∞. (10)Observe that if φ0,φ1 ∈ Ker Ac then φ(t) ∈ Ker Ac for all t . Similarly, if φ0 ∈ Ker Ac and φ1 = (0,0), then φ(t) = φ0for all t > 0.The orthogonal of the subspace Ker Ac in H is the following:H⊥c ={v = (c−1ψ,1,−ψ) ∈ H, ψ ∈ H 1(ω)}. (11)The subspace H⊥c is generated by v0 and vk , k  1. We denote with Vc,⊥ the orthogonal in V of Ker Ac; it is alsogenerated, in V, by v0 and vk , k  1.Let us remark that the energy,E(t,φ) = 12{∥∥φ′(t)∥∥2H +∫ω(φ1,1(ξ) + cφ2(ξ))2 dξ}= E(0,φ) = 12{∥∥φ1∥∥2H +∫ω(φ01,1(ξ) + cφ02(ξ))2 dξ} ∀t ∈ (0, T ), (12)defines a norm over Vc,⊥ × H⊥c . Moreover if (φ0,φ1) ∈ Vc,⊥ × H⊥c then φ(t) ∈ Vc,⊥ for all t .632 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–6342.2. Observability inequalityLet us recall that B is a normal system of boundary conditions when there exists a complementary system C suchthat {B,C} are the reduced Cauchy data of A; in our particular situation Cφ = φ1,1 + cφ2. Since the control is onlyactive at ξ = 1, the control property of the system (1) is related, see [4], to the existence of two positive constants C1and C2 such that, for all (φ0,φ1) ∈ V × H and T > 0 large enough the solution of (3) satisfies:C1∥∥(φ0,φ1)∥∥2V ×H T∫0(Cφ(1, t))2dt  C2∥∥(φ0,φ1)∥∥2V ×H . (13)Since 0 ∈ σ(Ac), the left inequality (called the observability inequality) cannot hold for all (φ0,φ1) ∈ V × H . Itsuffices to take φ0,φ1 ∈ Ker Ac so that φ(t) ∈ Ker Ac for all t and Cφ(1, t) = 0.Proposition 2.1. Let c > 0 and γ (c) = min(2c,√c2 + π2 − c). For all time T > T (c) ≡ 2π/γ (c), there exist twostrictly positive constants C1(c) and C2(c) such that (13) holds for all (φ0,φ1) ∈ Ṽ0,⊥ × H̃⊥0 .Proof of Proposition 2.1. When (φ0,φ1) ∈ Ṽ0,⊥ × H̃⊥0 (13) is equivalent to,C1(c)E(0,φ) T∫0(φ1,1(1, t) + cφ2(1, t))2 dt  C2(c)E(0,φ). (14)As usual in this context, these inequalities may be obtained from a direct application of a Ingham’s theorem on non-harmonic series (see e.g. [6, p. 59]). Indeed, with an easy computation one finds:E(0,φ) = c22(A20 + B20) + 12∞∑k=1λk(A2k + B2k). (15)On the other hand, one has:φ1,1(1, t) + cφ2(1, t) = c(A0 cos(μ0t) + B0 sin(μ0t)) + √2 ∞∑k=1(−1)kμ2k(Ak cos(μkt) + Bk sin(μkt))= 1√2∞∑k=1(−1)kμ2k(Ak + iBk) e−iμkt +c2(A0 + iB0) e−iμ0t+ c2(A0 − iB0) eiμ0t + 1√2∞∑k=1(−1)kμ2k(Ak − iBk) eiμkt . (16)We then apply the previously quoted Ingham’s theorem with I = (0, T ) and the sequence,w = (. . . ,−μ2,−μ1,−μ0,μ0,μ1,μ2, . . .),to obtain that the existence of two positives constants C1(c) and C2(c) such that (14) holds for all c > 0, under thecondition T > 2π/γ , with γ = min(μ0 − (−μ0), infk∈N |μk −μk−1|). From the concavity of the square root function,we deduce that |μ1 − μ0|  |μk+1 − μk| for all k  0 and then that γ = γ (c). The lower bound value T  of observability time is reached when c = π2/8 for which T  = √8. We observe thatthe time of controllability T  blows up as c goes to zero. This is due to the eigenvalue λ0 which vanishes with c.Precisely, if the initial condition is (φ0,φ1) = (v0,μ0v0) so that φ1,1(1, t) + cφ2(1, t) = c(cos(μ0t) + sin(μ0t)) thenwe obtain explicitly that the constant C1(c) goes to zero as c goes to zero unless T = O(c−1). Consequently, theobservability inequality is not uniform with respect to c. If we denote with HK , resp. VK , the closed subspace of H,resp. V, generated by vk for all k  1, we have the following result:F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–634 633Proposition 2.2 (Uniform observability w.r.t. c). Let c > 0 and γ (c) = √c2 + 4π2 − √c2 + π2. For allT > T (c) ≡ 2π/γ (c), there exist two positive constants C1 and C2 independent of c such that (13) holds forall (φ0,φ1) ∈ VK × HK .The lower bound T  is now a monotonous increasing function of c such that limc→0 T (c) = 2, lower bound forthe wave equation controlled at one extremity. We also remark that 2 < T (c) < T (c) for all c > 0.3. Exact controllabilityWe now apply the Hilbert Uniqueness Method of J.-L. Lions [7]. Let be V ′ = H−1(ω)×L2(ω) so that one has theusual situation V ⊂ H ≈ (H )′ ⊂ V ′ with H as pivot space. Let us assume that (y0,y1) ∈ H ×V ′. Formal integrationsby part provide that v is a control for the system (1) if and only ifT∫0(φ1,1 + cφ2)(1, t)v(t)dt =〈φ0,y1〉V ,V ′ −(φ1,y0)H (17)for every φ solution of (3). We then introduce the continuous and convex functional Jc :V × H → R defined by:Jc(φ0,φ1) = 12T∫0(φ1,1 + cφ2)2(1, t)dt −〈φ0,y1〉V ,V ′ +(φ1,y0)H. (18)If Jc is coercive, then Jc admits a unique minimum and the HUM control of minimal L2(0, T )-norm is given byvc = (φ1,1 + cφ3)(1, ·). Let us remark at first that Jc is only coercive on Ṽ0,⊥ × H̃⊥0 (provided T be large enough).Furthermore, if (y0,y1) belongs to Ker Ac, then 〈φ0,y1〉V ,V ′ − (φ1,y0)H = 0 for all (φ0,φ1) ∈ Ṽ0⊥ × H̃⊥0 andfrom (17), the control is zero; in this case, the solution y remains in Ker Ac for all t > 0 but is not controlled! Hencewe take y0 ∈ H⊥c and y1 ∈ (Ṽ0)⊥ where (Ṽ0)⊥ ⊂ V ′ is the orthogonal of Ṽ0 in the duality 〈·, ·〉V ,V ′ and we have:Theorem 3.1. Let c > 0. For any T > T (c) and any initial data (y0,y1) ∈ H⊥c × Ṽ⊥0 , there exists a control functionv ∈ L2(0, T ) which drives to rest at time T the solution y of (1) associated with (y0,y1). Moreover, the control ofminimal L2-norm is given by v(t) = (φ1,1 + cφ2)(1, t), where φ is the solution of (3) with (φ0,φ1) minimum of Jcdefined by (18) over Ṽ0,⊥ × H̃⊥0 .Let us explicitly remark that Ṽ⊥0 is the closure of Ṽ0,⊥ in V′. From Proposition 2.2, one has the following, whereV̂K is the closure of VK in V ′:Theorem 3.2 (Uniform controllability w.r.t. c). Let c > 0. For any T > T (c) and any initial data (y0,y1) ∈HK × V̂K , there exists a control function v ∈ L2(0, T ) which drives to rest at time T the solution yc of (1) asso-ciated with (y0,y1). Moreover, the control of minimal L2-norm is given by vc(t) = (φ(c)1,1 + cφ(c)2)(1, t) whereφ(c) is solution of (3) with (φ0,φ1) minimum of Jc defined by (18) over VK, × HK . At last, this control convergesweakly in L2(0, T ) as c goes to zero toward the control of minimal L2-norm which drives to rest the solution of thewave equation associated with (y01 , y11).4. Concluding remarks1. System (Sc) is, for c > 0, null controllable by a Dirichlet force only when the initial data (y0,y1) are in H⊥c ×Ṽ⊥0 .Let us point out that the subspaces H⊥c of H and Ṽ⊥0 of V′ are spanned by the eigenfunctions associated to the sequenceof eigenvalues λk whose elements have the following separation property:(Σ) there exists λ∗ > 0 such that dist({λk}k0, σess(Ac)) = λ∗ and lim λk = +∞. (19)k→∞634 F. Ammar-Khodja et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 629–6342. The analysis also reveals that the time of controllability increases when c goes to zero and that there exists a non-controllable mode for c = 0. Also in this case the subspace of controllable initial data is spanned by the eigenfunctionsassociated to the eigenvalues satisfying a condition of type (Σ ), [1].3. Analogous conclusions are true for a Neumann control Cy = y1,1 + cy2 at one extremity, [1].4. In [2] and [3] it has been proved that the exact controllability problem for spherical membrane fails when thevibrations are due to external actions of wavelengths connected to the eigenvalues with finite accumulation point(this point is exactly the essential spectrum of the operator). One can prove that the subspace of controllable initialdata also in this case is spanned by the eigenfunctions associated with the sequence of eigenvalues satisfying thecondition (Σ ), [1].5. The characterization of the initial conditions that are controllable for the membrane shell that have essentialspectrum, see [8], is open.References[1] F. Ammar-Khodja, G. Geymonat, A. Münch, in preparation.[2] G. Geymonat, P. Loreti, V. Valente, Exact controllability of thin elastic hemispherical shell via harmonic analysis, in: Boundary Value Problemsfor Partial Differential Equations and Applications, Masson, 1993.[3] G. Geymonat, P. Loreti, V. Valente, Spectral problems for thin shells and exact controllability, in: Spectral Analysis of Complex Structures, in:Travaux en cours, vol. 49, Hermann, Paris, 1995, pp. 35–57.[4] G. Geymonat, V. Valente, A noncontrollability result for systems of mixed order, SIAM J. Control Optim 39 (3) (2000) 661–672.[5] G. Grubb, G. Geymonat, The essential spectrum of elliptic boundary value problem, Math. Ann. 227 (1977) 247–276.[6] V. Komornik, P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, 2004.[7] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisations de systèmes distribués, 2 tomes, Masson, Paris, 1988.[8] J. Sanchez Hubert, E. Sanchez-Palencia, Coques élastiques minces : propriétés asymptotiques, Masson, Paris, 1997.

On the exact controllability of a system of mixed order with essential spectrum

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On the exact controllability of a system of mixed order with essential spectrum

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