Insensitizing controls for a nonlinear parabolic equation with a nonlinear term involving the state and the gradient in unbounded domains. (English) Zbl 1171.93009
Summary: We consider the existence of insensitizing control for a semilinear heat equation involving gradient terms in unbounded domain \(\Omega\). In this case, we prove the existence of controls insensitizing the \(L^2\)-norm of the observation of the solution in an open subset of the domain. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.
MSC:
| 93B05 | Controllability |
| 35K55 | Nonlinear parabolic equations |
| 35K05 | Heat equation |
| 93C20 | Control/observation systems governed by partial differential equations |
References:
| [1] | J.-L. Lions, Quelques notions dans l’analyse et le contrôle de systèmes à données incomplètes. In: Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics. Málaga, University of Málaga, 1990, pp. 43-54; J.-L. Lions, Quelques notions dans l’analyse et le contrôle de systèmes à données incomplètes. In: Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics. Málaga, University of Málaga, 1990, pp. 43-54 |
| [2] | Bodart, O.; Gonzaáez-Burgos, M.; Peéez-Garca, R., Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., 57, 5-6, 687-711 (2004) · Zbl 1055.35022 |
| [3] | L. De Teresa, Controls insensitizing the norm of the solution of a semilinear heat equation in unbounded domains, ESAIM: COCV, vol. 2, pp. 125-149, 1997; L. De Teresa, Controls insensitizing the norm of the solution of a semilinear heat equation in unbounded domains, ESAIM: COCV, vol. 2, pp. 125-149, 1997 · Zbl 0895.93023 |
| [4] | De Teresa, L., Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25, 1-2, 39-72 (2000) · Zbl 0942.35028 |
| [5] | De Teresa, L.; Zuazua, E., Approximate controllability of the heat equation in unbounded domains, Nonlinear Anal. TMA, 37, 1059-1090 (1999) · Zbl 0946.93006 |
| [6] | Cabanillas, V.; De Menezes, S.; Zuazua, E., Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optim. Theory Appl., 110, 2, 245-264 (2001) · Zbl 0997.93048 |
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