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The optimal stochastic control problem with a quadratic cost functional forlinear partial differential equations (PDEs) driven by a state-andcontrol-dependent white noise is formulated and studied. Both finite-andinfinite-time horizons are considered. The multi-plicative white noise dynamicsof the system give rise to a new phenomenon of singularity to the associatedRiccati equation and even to the Lyapunov equation. Well-posedness of bothRiccati equation and Lyapunov equation are obtained for the first time. Thelinear feedback coefficient of the optimal control turns out to be singular andexpressed in terms of the solution of the associated Riccati equation. The nullcontrollability is shown to be equivalent to the existence of the solution toRiccati equation with the singular terminal value. Finally, the controlledAnderson model is addressed as an illustrating example.

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