Inégalité d'observabilité du type logarithmique et estimation de la fonction de coût des solutions des équations hyperboliques

Ouksel, Leila

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 318-342

Dans ce travail, nous donnons une estimation logarithmique des données de la solution $u$, d’un problème hyperbolique avec condition aux limites de type Neumann, par la trace de $u$ restreinte à un ouvert du bord, pendant un temps suffisamment grand qui nous permet d’estimer la fonction de coût de ce problème.

This work proposes a logarithmic estimation of the initial values of the solution $u$ of a hyperbolic problem, with Neumann boundary conditions, using the trace of $u$ restricted to the neighbourhood of the boundary, during a time sufficiently large for estimating the cost function of the problem.

Classification : 35B37, 93B05, 93B07, 35L20
Mots-clés : problème hyberbolique, contrôle, fonction de coût, inégalité de Carleman

[1] S. Alinhac, Non unicité du problème de Cauchy. Ann. Math 117 (1983) 77-108. | Zbl | MR

[2] S. Alinhac and M.S. Baouendi, A non uniqueness result for operators of principal type. Math. Z 220 (1995) 561-568. | Zbl | MR

[3] H. Bahouri, Dépendence non linéaire des données de Cauchy pour des solutions des équations aux dérivées partielles. J. Math. Pures. Appl 66 (1987) 127-138. | Zbl | MR

[4] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim 30 (1992) 1024-1065. | Zbl | MR

[5] N. Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes. Mém. Soc. Math. France (N. S.) 55, Marseilles (1993). | Zbl | Numdam

[6] T. Duyckaerts, Optimal decay rates of the energy of an hyperbolic-parabolic system coupled by an interface. European Union Projects “Smart System” (2002).

[7] T. Duyckaerts, Xu Zhang and E. Zuazua, On the optimality of the observability inequality for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear). | MR | Numdam

[8] C. Fabre, Résultats de contrôlabilité exacte interne pour l'équation de Schrödinger et leurs limites asymptotiques : Application à certaines équations de plaques vibrantes. Asym. Anal 5 (1992) 343-379. | Zbl | MR

[9] E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Diff. Equa 5 (2000) 465-514. | Zbl | MR

[10] A.V. Fursikov and O. Yu Imanivilov, Controllability of evolution equations. Lect. Notes Ser 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | Zbl | MR

[11] L. Hörmander, Linear partiel differential operators. Springer-Verlag, Berlin (1963). | Zbl

[12] L. Hörmander, On the uniqueness of the Cauchy problem under partial analyticity asumptions. Springer-Verlag (1996). | Zbl | MR

[13] V.-M. Isakov, On the uniquenss of the solution of the Cauchy problem. Sov. Math. Dokl 22 (1980) 639-642. | Zbl | MR

[14] F. John, Continous dependence on data for solution of partial differential equations with prescribed bound. Comm. Pure. Appl. Math 17 (1960) 551-585. | Zbl | MR

[15] I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilisation of Schrödinger equation with Dirichlet control. Diff. Integral. Equa 5 (1992) 521-535. | Zbl | MR

[16] G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures. Appl 71 (1992) 267-291. | Zbl

[17] G. Lebeau, Contrôle analytique I : estimation a priori. Duk. Math. J 68 (1992) 1-30. | Zbl | MR

[18] G. Lebeau et L. Robbiano, Contrôle exacte de l'équation de la chaleur. Comm Partial. Diff. Equa 20 (1995) 335-356. | Zbl | MR

[19] G. Lebeau et L. Robbiano, Stabilisation de l'équation des ondes par le bord Duk. Math. J. 86 (1997) 465-491. | Zbl | MR

[20] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués. Masson Collection, RMA, Paris (1988). | Zbl

[21] J.-L. Lions et E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Dunod, Paris (1968). | Zbl | MR

[22] E. Machtyngier, Exact controllability for Schrödinger equation. SIAM J. Control. Optim 32 (1994) 24-34. | Zbl | MR

[23] L. Miller, Geometric bounds on the growth rate of null-controllability cost of the heat equation in small time. J. Diff. Eq. 2004 (2004) 202-226. | Zbl | MR

[24] L. Miller, On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math 129 (2005) 175-185. | Zbl | MR

[25] K.-D. Phung, Observability and control of Schrödinger equation. SIAM J. Control Optim 40 (2001) 211-230. | Zbl | MR

[26] K.-D. Phung, Note on the cost of the approximate controllability for the heat equation with potentiel. J. Math. Anal. Appl. 295 (2004) 527-538. | Zbl | MR

[27] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures. Appl 84 (2005) 407-470. | Zbl | MR

[28] L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques. Comm Partial. Diff. Equa 16 (1991) 789-800. | Zbl

[29] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques Asym. Anal. 10 (1995) 95-115. | Zbl | MR

[30] L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operator with partially holomorphic coefficients. Inventiones Mathematice 131 (1998) 493-539. | Zbl | MR

[31] D. Russell, A unified boundary contrllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math 52 (1973) 189-212. | Zbl | MR

[32] D. Tataru, Unique continuation for solution to P.D.E's between Hörmander theorem and Holmgren's theorem Comm. Part. Diff. Eq 20 (1995) 855-884. | Zbl | MR

[33] D. Tataru, Carleman estimates and unique continuation for solutions to boundary-value problems J. Math. Pures. Appl 75 (1996) 367-408. | Zbl | MR

[34] D. Tataru, Unique continuation for partial differential operators with partially analytic coefficients J. Math. Pures. Appl 78 (1999) 505-521. | Zbl | MR

[35] X. Zhang and E. Zuazua, Polinomial decay and control of a 1-d model for fluid-structure interaction. C. R. Acad. Sci., Paris, Ser. I 336 (2003) 745-750. | Zbl | MR

[36] X. Zhang and E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system. C. R. Acad. Sci., Paris, Ser. I 336 (2003) 823-828. | Zbl | MR

[37] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal 184 (2007) 49-120. | MR