Optimal control of stochastic systems with memory under noisy observations

Abstract

This thesis aims to study a new type of stochastic partial differential equations (SPDEs) with space interactions. By space interactions, we mean that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. In the first part, we introduce linear SPDEs. Then we prove the existence and uniqueness results (mild solution) for nonlinear SPDEs under linear growth and Lipschitz conditions on the coefficients. In the second part of this thesis, using results from Noisy Observation (nonlinear filtering), we transformed this noisy observation stochastic differential equation (SDE) control problem into full observation stochastic partial differential equations (SPDEs), and then we prove a sufficient and necessary maximum principle for the optimal control of SPDEs. In the third part of this thesis, we prove the existence and uniqueness of strong, smooth solutions of a class of stochastic partial differential equations with space interactions., and we show that, under some conditions,we usewhite noise theory to prove a positivity theorem for a class of SPDEs with space interactions. The solutions are positive for all times if the initial values are. Then we study the general optimization problem for such a system. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an example of optimal vaccination strategy for epidemics modelled as stochastic partial differential equations (SPDEs) with space interactions.