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

Abstract

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.