Adaptive Stabilization of Noncooperative Stochastic Differential Games

.In this paper, we consider the adaptive stabilization problem for a basic class of linear-quadratic noncooperative stochastic differential games when the systems matrices are unknown to the regulator and the players. This is a typical problem of game-based control systems (GBCS) introduced and studied recently, which have a hierarchical decision-making structure: there is a controller at the upper level acting as a global regulator which makes its decision first, and the players at the lower level are assumed to play a typical zero-sum differential games. The main purpose of the paper is to study how the adaptive regulator can be designed to make the GBCS globally stable and at the same time to ensure a Nash equilibrium reached by the players, where the adaptive strategies of the players are assumed to be constructed based on the standard least squares estimators. The design of the global regulator is an integration of the weighted least squares parameter estimator, random regularization and diminishing excitation methods. Under the assumption that the system matrix pair \((A, B)\) is controllable and there exists a stabilizing solution for the corresponding algebraic Riccati equation, it is shown that the closed-loop adaptive GBCS will be globally stable, and at the same time reach a Nash equilibrium by the players.Keywordsstochastic adaptive systemszero-sum dynamical gamesparameter estimationgame-based control systemsNash equilibriumrandom excitationMSC codes93D93E