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

TL;DR

This paper studies stochastic control problems for PDEs with state-and control-dependent white noise, revealing new singularities in Riccati equations and establishing well-posedness results for these equations in finite and infinite horizons.

Contribution

It introduces the first well-posedness results for Riccati and Lyapunov equations with singularities caused by multiplicative white noise in PDE control problems.

Findings

Well-posedness of Riccati and Lyapunov equations with singularities is established.

The optimal control feedback law is expressed via the Riccati equation solution.

Null controllability is linked to the existence of solutions with singular terminal conditions.

Abstract

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.