STOCHASTIC LQ CONTROL 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 PDEs driven by a state-and control-dependent white noise is formulated and studied. Both finite and infinite-time horizons are considered. The multiplicative 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 the Riccati equation and the 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 the Riccati equation with the singular terminal value. Finally, the controlled Anderson model is addressed as an illustrating example.