Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients

Let $H$ and $U$ be real Hilbert spaces, $A^1\colon D(A^1)\subset H\rightarrow H$ the generator of a $C_0$-semigroup in $L(H)$, and $W$ a cylindrical Brownian motion, and let $A^2,B,C$ and $S$ be bounded $\bold{F}^W$-adapted random processes with values in suitable spaces; moreover, $S_t$ is assumed to be a positive semidefinite operator, for all $t\ge 0.$ The authors of the present paper study the infinite horizon stochastic control problem composed of the linear infinite-dimensional stochastic state equation $dX_t^{x,u}=((A^1+A^2_t)X_t+B_tu_t)dt+C_tX_tdW_t,$ $X_0^{x,u}=x\in H$ ($u$ is a $U$-valued progressively measurable control process) and of the quadratic cost functional $J(0,x;u)={\Bbb E\int_0^{+\infty}((S_tX_t,X_t)+\vert u_t\vert^2)dt.}$ While linear-quadratic control problems with deterministic coefficients have been studied in the infinite-dimensional case by several authors, in the case of random coefficients such a problem has been studied up to now only for finite horizon problems (L. Gr\"{u}ne, 1998). In the context studied by the authors, the associated Riccati equation is an infinite time horizon backward stochastic differential equation that involves quadratic nonlinearities and takes its values in the non-Hilbert space $L(H)$. To cope with the difficulty met by Grüne (1998), the authors propose a notion of generalized solution of the BSDE. They prove the existence of a minimal non-negative solution and, under appropriate assumptions, its uniqueness. Moreover, they show that the solution of the Riccati equation allows them to perform the synthesis of the optimal control for the above control problem, and they study its attractivity and maximality properties. Finally, they also investigate special features of the stationary case: They prove, in particular, that the stationarity of the coefficients involves that of the minimal non-negative solution of the Riccati equation. The authors show how their results apply to the problem of the linear-quadratic control of an infinite time horizon wave equation in random media.