An inverse problem of finding a time-dependent parameter in a bilinear heat equation

Abstract This paper investigates the inverse problem of estimating an unknown bilinear control u , applied locally in a reaction-diffusion equation. Our goal is to achieve a temperature profile close to a desired reference at the final time. We formulate the problem as an optimal control framework and analyze the existence, optimality conditions and stability of the solution with respect to the data. An algorithm and some numerical experiments are proposed to show the effectiveness of our approach in steering the system towards a desired state.