Alguns resultados de controle para equações diferenciais ordinárias lineares

Abstract

In this work, we deal with some control results for systems of ordinary dierential equations. We are interested in knowing if it is possible to inuence the behavior of the solutions of a given equation in such a way that they behave the way we want in a nite time. This intuitive notion brings us the concept of an accessible state set, the one consisting of data that are possible to be reached in a nite time, where we study the topological properties of this set. We will see that there is a quite interesting distinction when we study problems with or without restrictions, where we present some controllability results in each case. Also, we will see the optimization problem concerning the time, that is, we will answer the question related to the existence of control taking one data to another in the minimal time possible. Finally, we study the Linear-Quadratic problem (L.Q.), where our goal is to know which is the best control that minimizes a given cost associated with the trajectories of the system and the control itself. To this class of problems, we refer, usually, as optimal control problems. At this point, we will see that the optimal solutions of an open-loop system can be written as the optimal solutions of a closed-loop system. In order to do that, we will make use of the Riccati equation.