Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise

The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state-and control-dependent white noise is formulated and studied. Both finite-and infinite-time horizons are considered. The multi-plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular and expressed in terms of the solution of the associated Riccati equation. The null controllability is shown to be equivalent to the existence of the solution to Riccati equation with the singular terminal value. Finally, the controlled Anderson model is addressed as an illustrating example.