Feedback null controllability of the semilinear heat equation

Abstract

In this paper we study the null controllability of the semilinear heat equation studying the properties of the minimum energy that one needs to steer an initial state $x$ in $0$. We prove this is locally Lipschitz, and consequently we obtain the expected optimal feedback law. We also characterize the value function as the unique positive viscosity solution (of the corresponding Hamilton--Jacobi equation with singular final data) which tends to $0$ on admissible trajectories, or as the minimal positive viscosity supersolution.