针对噪声同时依赖于状态与 控制的~It\^{o} 型离散~Markov 切换系统的随机零和博弈问题,分别讨论了其在有限时域和无限时域情形的鞍点均衡策略.利用配方法,得到了随机零和博弈问题存在解的充分必要条件等价于相应的耦合~Riccati 差分~(代数) 方程存在解,并给出了最优解的显式形式.然后把所得的结果应用于相应的~$H_{\mathrm{\infty }}$ 控制问题,得到了~$H_{\mathrm{\infty }}$ 控制策略的~Riccati 方程,最后 给出了数值算例.

The linear quadratic stochastic zero-sum games for discrete-time Markov jump systems with state and control-dependent noise are discussed in this paper. By the~square~completion~technique, it is shown that the necessary and sufficient conditions for the existence of stochastic zero-sum games is equivalent to the solvability of the associated cross-coupled Riccati algebraic equations. Moreover, the explicit expressions of the optimal strategies are constructed, and the results are applied to $H_{\mathrm{\infty }}$ control problem for discrete-time Markov jump systems. Moreover, an illustrative example is also proposed.