Feedback Null Controllability Of The Semilinear Heat Equation

In this paper we consider the quadratic control problem for the semilinear heat equation with Lipschitz nonlinearityand associated cost function. We prove that the value function is locally Lipschitz using observability inequalities and also characterize it as the unique positive viscosity solution of the corresponding Hamilton-Jacobi equation. Consequently we obtain the expected optimal feedback law.The paper is organized into three chapters. First, the method we use and the relative recent research results are introduced briefly in chapter 1. In chapter 2, we prove that for fixed 0 ≤ t < T, φ(·, x) is locally Lipschitz and consequently we obtain the expected optimal feedback law. The main results are as follows:Theorem 2.1 For each 0 ≤ t < T, φ(t, ·) is locally Lipschitz . Moreover, if (u*,y*) is such a pair that the infimum is attained in the value function thenChapter 3 is devoted to discuss the value function as the minimal positive viscosity supersolution of (1.6) subject to (1.7). In particular the existence of a positive viscosity supersolution is equivalent to the null controllability of the state system. The main results are as follows:Theorem 3.1 (i) If there exists ip e C([0,T] x H) a positive viscosity supersoluion of (3.1) satisfying the final condition (3.7), (3.8) and (3.9), then the state system (1.2) is null controllable and .(ii) If the state system (1.2) is null controllable and (t,-) is locally Lipschitz for each t