Optimal control of stochastic partial differential equations in Banach spaces

In this thesis we study optimal control problems in Banach spaces for stochastic partial differential equations. We investigate two different approaches. In the first part we study Hamilton-Jacobi-Bellman equations (HJB) in Banach spaces associated with optimal feedback control of a class of non-autonomous semilinear stochastic evolution equations driven by additive noise. We prove the existence and uniqueness of mild solutions to HJB equations using the smoothing property of the transition evolution operator associated with the linearized stochastic equation. In the second part we study an optimal relaxed control problem for a class of autonomous semilinear stochastic stochastic PDEs on Banach spaces driven by multiplicative noise. The state equation is controlled through the nonlinear part of the drift coefficient and satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.