Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

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

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Abstract In this paper we consider a semilinear heat equation (in a bounded domain Ω of ℝ N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ⊂ Ω, that insensitizes the L 2 − norm of the observation of the solution in another open subset 𝒪 ⊂ Ω when ω ∩ 𝒪 ≠ ∅, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L r -controls (r large enough) starting from insensitizing controls in L 2.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/edce2229259260eeb49a6e28f46d4fe2eadb2a4a

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Citations in the corpus

RECENT RESULTS ON THE CONTROLLABILITY OF LINEAR COUPLED PARABOLIC PROBLEMS: A SURVEY (2011, methodology): “We will use some ideas from [14] and [40].”
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]).”
Controllability results for cascade systems of m coupled parabolic PDEs by one control force (2010, unknown): “In [19] and [5] the authors consider a nonlinear system of two heat equations, one of them being forward and the other one”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, result): “Interesting results in this direction have been proved in [44], [13], [62], in particular.”