Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

By Olivier Bodart, Manuel González-Burgos, Rosario Pérez-García. 2004

63 citations recorded by the source corpus.

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of IR N . We rst consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F (y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2 norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate xed point arguments.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f0d4a3b288e443cbad257511967cf878516c825d

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Citations in the corpus

RECENT RESULTS ON THE CONTROLLABILITY OF LINEAR COUPLED PARABOLIC PROBLEMS: A SURVEY (2011, result): “1 we have adapted the proof of a similar result given in [63] (see also [15]) to the one-dimensional case.”
CONTROLLABILITY OF 1-D COUPLED DEGENERATE PARABOLIC EQUATIONS (2009, background): “The controllability properties of nondegenerate parabolic cascade systems have been studied in different contexts in the last fifteen years or so (see [2, 22, 3, 4, 14, 16, 18]).”
New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence (2016, background): “The first results on controllability of coupled parabolic equations (n > 1) have been established in [35, 13, 3, 25].”
Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity (2004, methodology): “In Bodart et al. (2004), the authors prove a result on the existence of insensitizing controls for a semilinear heat equation with a nonlinear term fðy;HyÞ, with f : R RN 7!”
Controllability to trajectories for some parabolic systems of three and two equations by one control force (2013, unknown): “To the best of our knowledge most of the existing results in the litterature deal with zero order coupling systems or constant coefficients (see for instance [29], [1] [9], [2], [3], [13], [19], [21], [4], [5], [8] and [20]).”
Boundary controllability of parabolic coupled equations (2010, background): “To our knowledge, all them deal with distributed controls, exerted on a small open set ω; see for instance [23], [6], [2], [5], [13], [14], [15], [3] and [4].”