Infinite horizon LQ Nash Games for SDEs with infinite jumps

Abstract

In this paper, we consider infinite horizon linear-quadratic (LQ) Nash games for stochastic differential equations (SDEs) with infinite Markovian jumps and (x, u, v)-dependent noise. An indefinite stochastic LQ result is first derived for the considered system. Then, under the condition of strong detectability, a necessary and sufficient condition for the existence of a Nash equilibrium is put forward in terms of the solvability of a countably infinite set of coupled generalized algebraic Riccati equations (ICGAREs). Moreover, the mixed H 2 ∕H ∞ control is investigated by Nash game approach as an important application. At last, we present an iterative algorithm to solve the ICGAREs, and a numerical simulation is given to illustrate its efficiency.

Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications

“…If the infinite horizon cost function is concerned, it is much more challenging owing to the requirement of stabilization limitation for the closed-loop system. As discussed in [26], the infinite horizon LQ Nash game has been considered.…”

“…Remark 5. By comparing Theorem 1 with Theorem 2, surely the existence of Nash equilibrium points and the solvability of finite horizon H 2 /H ∞ control are not equivalent for system (1), which differs from the continuous-time case as described in [26]. The main cause of the inequivalence lies in that the condition H 1 (t, ς, Λ 1 ) > 0 of Equation (35) cannot meet the Nash equilibrium problem, namely, L T < γ is not equivalent to L T ≤ γ.…”

“…From a new perspective, the finite horizon mixed H 2 /H ∞ control is further investigated. On the one hand, this is an extension of the previous study from MJSSs [22], to IMJSSs; on the other, it is a discrete-time case of [26]. Concretely, the main work and contribution of this thesis is as follows.…”

“…As a consequence, the related research about IMJSSs is of critical importance. In the game field, [26] discussed an infinite horizon linear quadratic (LQ) Nash game for continuous-time IMJSSs. Unfortunately, to the best of our knowledge, there is almost no research about the Nash game problem for discrete-time IMJSSs.…”