Forward–backward linear–quadratic optimal control and stabilization problems for discrete-time stochastic delayed system

Abstract

This paper is concerned with a type of forward–backward linear–quadratic (FBLQ) stochastic optimal control and stabilization problems for the discrete-time forward–backward stochastic delayed system (FBSDS). We first present the necessary and sufficient unique solvability condition of a kind of general forward–backward stochastic difference delayed equations (FBSDDEs) in terms of the Riccati-ZXL difference equations. Then, by virtue of the solution to the FBSDDEs and the stochastic maximum principle, we explicitly design the optimal controller, which is a linear function of the conditional expectation of the state. In addition, an equivalent condition of the mean-square exponential stabilizability for the FBSDS is obtained by investigating the asymptotic behavior of the Riccati-ZXL difference equations. Finally, some numerical examples are exploited to illustrate the validity of the obtained results.