Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients
Linear quadratic optimal control problems (LQOCs) with stochastic coefficients and the associated backward stochastic Riccati equation (BSRE) have recently been studied by several authors. The authors of the present paper are the first to study this problem with stochastic coefficients, infinite-time horizon and an infinite-dimensional state space equation. As in the case of LQOCs with deterministic coefficients, a finite cost condition (stabilizability) has to be required by the authors; it is stated in form of the existence of an admissible stochastic control process for which the cost over all time intervals \([t,+\infty)\), \(t\in R\), is finite. With the help of a monotonicity argument the authors show the equivalence between stabilizability and the existence of a solution of the associated BSRE, where a solution is understood as generalized solution of the BSRE restricted to the finite time interval \([0,T]\), for all \(T>0\), as it has been introduced by \textit{G. Guatteri} and \textit{G. Tessitore} [SIAM J. Control Optim. 44, No. 1, 159--194 (2005; Zbl 1102.93041)]. Moreover, they show that if the stability condition is satisfied, a minimal non negative solution exists. It is the unique solution of the BSRE if the minimal non negative solution stabilizes the state equation with respect to the identity uniformly with respect to the time. The authors show that such a solution allows to perform the synthesis of the optimal control generalizing classical results holding in the deterministic coefficients case with the BSRE. They also investigate the attractivity properties of the unique solution of the BSRE. Finally, they study the case of stationary coefficients and propose an example concerning a controlled wave equation in random media.
scientific article; zbMATH DE number 5345687
scientific article