Controlabilidade nula e aproximada para uma equação parabólica não linear

Abstract

The Mathematical control theory is an area of applied mathematics that deals with the analysis of Partial Differential Equations (EDPs) or Ordinary Differential Equations (ODE) control systems. This theory had a great development with the works of Russel, J. Lions, O.Yu. Imanuvilov, A. V. Fursikov, E. Zuazua, among others. In this work we will study the null and approximate controllability of a parabolic equation: the nonlinear heat equation. Our proof is based on the fact that non-linearity is globally Lipschitz. Thus, we will demonstrate the existence of a control u in a space with weight that, when acting in the domain, leads the system to the state of equilibrium. We show that null controllability can be obtained through Carleman’s inequality, inequality of observability and using the arguments of the fixed-point theorem for a multi-value application. In the case of approximate controllability, using Carleman’s inequality, we show that the set of admissible states RL(T) is dense in L 2 (Ω).