Optimal control problems constrained by stochastic partial differential equations

Abstract

In this thesis, we solve several optimal control problems constrained by linear as well as nonlinear stochastic partial differential equations by a stochastic maximum principle. We provide some basic concepts from functional analysis and a stochastic calculus to obtain existence and uniqueness results of mild solutions to these equations. For the linear case, we consider two specific examples, where we involve nonhomogeneous boundary conditions using the theory of fractional powers of closed operators. First, we treat the stochastic heat equation with nonhomogeneous Neumann boundary conditions, where controls and additive noise terms appear inside the domain as well as on the boundary. Here, the control problem is described by tracking a desired state at the terminal point of time leading to a convex optimization problem. Using a stochastic maximum principle, we state necessary and sufficient optimality conditions, which we utilize to design explicit formulas for the optimal controls. By a reformulation of these formulas, we finally obtain a feedback law of the optimal controls. Next, we consider the stochastic Stokes equations with nonhomogeneous Dirichlet boundary conditions, where we include a linear multiplicative noise term. Here, controls appear inside the domain as well as on the boundary. The control problem is defined by tracking a desired state through the whole time interval leading to a convex optimization problem. Again, we state necessary and sufficient optimality conditions the optimal controls have to satisfy. The design of these optimal controls is mainly based on a duality principle giving relations between the mild solutions of forward equations and a backward equation. Here, the forward equations are given by the partial G^ateaux derivatives of the stochastic Stokes equations with respect to the controls and the backward equation is characterized by the adjoint equation. To derive this duality principle, an approximation of the mild solutions by strong solutions is required, which we obtain using the resolvent operator. This provides formulas for the optimal controls based on the adjoint equation. As a consequence, it remains to solve a system of coupled forward and backward stochastic partial differential equations. For the nonlinear case, we study the stochastic Navier-Stokes equations with homogeneous Dirichlet boundary conditions, where we include a linear multiplicative noise term. Here, the theory of fractional powers of closed operators gives a treatment of the convection term arising in these equations. In general, it is not possible to define a solution over an arbitrary time interval. We overcome this problem using a local mild solution well defined upto a certain stopping time. Hence, the cost functional related to the control problem has to incorporate this stopping time leading to a nonconvex optimization problem. Thus, a stochastic maximum principle provides only a necessary optimality condition. However, we still design the optimal control based on the adjoint equation using a duality principle. Again, it remains to solve a system of coupled forward and backward stochastic partial differential equations. Furthermore, we show that the optimal control satisfies a sufficient optimality condition based on the second order Fréchet derivative of the cost functional.