Uma introdução à controlabilidade de equações diferenciais parciais estocásticas

Abstract

In this work, we study the controllability of some Stochastic Differential Equations. First, we show some basic controllability results for this type of equations, such as the fact that stochastic evolution equations are not controllable when the control only acts on the deterministic part of the equation, thus requiring the application of a control in its stochastic part. Next, we analyze the controllability of two types of stochastic equations, namely the transport and heat equations. Using an observability estimate for reverse stochastic transport equations, we show that, for a sufficiently large time T > 0, the stochastic transport equation is exactly controllable. For the stochastic heat equation, we prove a global Carleman estimate for reverse stochastic parabolic equations and use this to obtain a unique continuation property and an observability estimate, which allow us to conclude the null and approximate controllability of the stochastic heat equation for any time T > 0.