Exact controllability for semilinear wave equations in one space dimension. (English) Zbl 0769.93017

Summary: The exact controllability of the semilinear wave equation \(y''- y_{xx}+f(y)=h\) in one space dimension with Dirichlet boundary conditions is studied. We prove that if \(| f(s)|/| s|\log^ 2| s|\to 0\) as \(| s|\to\infty\), then the exact controllability holds in \(H^ 1_ 0(\Omega)\times L^ 2(\Omega)\) with controls \(h\in L^ 2(\Omega\times(0,T))\) supported in any open and non-empty subset of \(\Omega\). The exact controllability time is that of the linear case where \(f=0\). Our method of proof is based on HUM (Hilbert Uniqueness Method) and on a fixed point technique. We also show that this result is almost optimal by proving that if \(f\) behaves like — \(s \log^ p(1+| s|)\) with \(p>2\) as \(| s|\to\infty\), then the system is not exactly controllable. This is due to blow-up phenomena. The method of proofs is rather general and applied also to the wave equation with Neumann type boundary conditions.


93B05 Controllability
35L05 Wave equation
93C20 Control/observation systems governed by partial differential equations
Full Text: DOI Numdam EuDML


[1] C. Bardos, G. Lebeau and J. Rauch, Contrôle et stabilisation dans les problèmes hyperboliques, Appendix II in J. L. Lions [L2], pp. 492-537.
[2] Bardos, C.; Lebeau, G.; Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, S.I.A.M. J. Control Optim., Vol. 30, 1024-1065, (1992) · Zbl 0786.93009
[3] Carmichael, N.; Quinn, M. D., Fixed point methods in nonlinear control, (Kappel, F.; Kunisch, K.; Schapacher, W., Distributed Parametes Systems, Lecture Notes in Control and Information Sciences, #75, (1985), Springer-Verlag Berlin), 24-51 · Zbl 0577.93028
[4] Cazenave, T.; Haraux, A., Equations d’évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse, Vol. 2, 21-51, (1980) · Zbl 0411.35051
[5] Chewning, W. C., Controllability of the nonlinear wave equation in several space variables, S.I.A.M. J. Control Optim., Vol. 14, 1, 19-25, (1976) · Zbl 0322.93009
[6] Cirina, M. A., Boundary controllability of nonlinear hyperbolic systems, S.I.A.M. J. Control, Vol. 7, 198-212, (1969) · Zbl 0182.20203
[7] Fattorini, H. O., Local controllability of a nonlinear wave equation, Math. Systems Theory, Vol. 9, 363-366, (1975) · Zbl 0319.93009
[8] Hermes, H., Controllability and the singular problem, S.I.A.M. J. Control, Vol. 2, 241-260, (1965) · Zbl 0163.10803
[9] I. Lasiecka and R. Triggiani, Exact Controllability of Semilinear Abstract Systems with Applications to Wave and Plates Boundary Control Problems, Proceedings of the 28th I.E.E.E. Conference on Decision and Control, 1989, pp. 2291-2294.
[10] Lasiecka, I.; Triggiani, R., Exact controllability of semilinear abstract systems with applications to wave and plates boundary control problems, Appl. Math. & Optim., Vol. 23, 109-154, (1991) · Zbl 0729.93023
[11] Lee, E. B.; Markus, L. W., Foundations of optimal control theory, (1967), John Wiley New York · Zbl 0159.13201
[12] Lions, J. L., Contrôlabilité exacte de systèmes distribués, C. R. Acad. Sci. Paris, Vol. 302, 471-475, (1986) · Zbl 0589.49022
[13] Lions, J. L., Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. tome 1. contrôlabilité exacte, (1988), Masson, RMA8 · Zbl 0653.93002
[14] Lions, J. L., Exact controllability, stabilization and perturbations for distributed systems, S.I.A.M. Rev., Vol. 30, 1-68, (1988) · Zbl 0644.49028
[15] Lukes, D. L., Global controllability of nonlinear systems, S.I.A.M. J. Control, Vol. 10, 112-126, (1972) · Zbl 0264.93004
[16] Markus, L., Controllability of nonlinear processes, S.I.A.M. J. Control, Vol. 3, 78-90, (1965) · Zbl 0294.93001
[17] Naito, K., Controllability of semilinear control systems, S.I.A.M. J. Control Optim., Vol. 25, 715-722, (1987) · Zbl 0617.93004
[18] Naito, K.; Seidman, Th. I., Invariance of the approximately reachable set under nonlinear perturbations, S.I.A.M. J. Cont. Optim., Vol. 29, 731-750, (1991) · Zbl 0729.49022
[19] Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, S.I.A.M. Rev., Vol. 20, 639-739, (1978) · Zbl 0397.93001
[20] Seidman, Th. I., Invariance of the reachable set under nonlinear perturbations, S.I.A.M. J. Control Optim., Vol. 23, 1173-1191, (1987) · Zbl 0626.49018
[21] Simon, J., Compact sets in the space L^{P}(0, T; B), Annali di Matematica pura ed Applicata, Vol. CXLVI, 65-96, (1987) · Zbl 0629.46031
[22] Zuazua, E., Contrôlabilité exacte de systèmes d’évolution non linéaires, C. R. Acad. Sci. Paris, Vol. 306, 129-132, (1988) · Zbl 0639.49029
[23] Zuazua, E., Exact controllability for the semilinear wave equation, J. Math. pures et appl., Vol. 69, 1-32, (1990) · Zbl 0638.49017
[24] Zuazua, E., An introduction to the exact controllability for distributed systems, Textos e Notas, 44, (1990), Universidades de Lisboa, C.M.A.F.
[25] E. Zuazua, Exact Controllability of Semilinear Distributed Systems, Proceedings of the 27th I.E.E.E. Conference on Decision and Control, 1988, pp. 1265-1268.
[26] Zuazua, E., Exact boundary controllability for the semilinear wave equation, in nonlinear partial differential equations and their applications, (Brezis, H.; Lions, J. J., Research Notes in Mathematics, X, (1991), Pitman), 357-391, Séminaire du Collège de France 1987/1988
[27] Zuazua, E., Contrôlabilité exacte d’une équation des ondes surlinéaire à une dimension d’espace, C. R. Acad. Sci. Paris, 311, 285-290, (1990) · Zbl 0721.93009
[28] Zuazua, E., Exponential decay for semilinear wave equations with localized damping, Comm. in P.D.E., Vol. 15, 2, 205-235, (1990) · Zbl 0716.35010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.