Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients
type: Item
instance of: scholarly article
title: Backward stochastic Riccati equations and infinite horizon L-Q optimal control with infinite dimensional state space and random coefficients
zbMATH Open document ID: 1157.60058
author: Giuseppina Guatteri
author: Gianmario Tessitore
published in: Applied Mathematics and Optimization
publication date: 2008-09-22T00:00:00Z
review text: 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.
reviewed by: Rainer Buckdahn
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 5345687
zbMATH Keywords: infinite horizon control problem
zbMATH Keywords: LQ optimal control problem
zbMATH Keywords: backward stochastic Riccati equation
zbMATH Keywords: stochastic coefficient
MaRDI profile type: Publication
full work available at URL: s00245-007-9020-y
OpenAlex ID: W1978411015
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DOI: 10.1007/S00245-007-9020-Y
description: scientific article; zbMATH DE number 5345687