Nash Differential Game for Discrete-time Markov Jump System with Partially Unknown Transition Probabilities

It is noted that transition probability matrix elements cannot be fully known. How to study Nash differential game for discrete-time Markov jump system (MJS) under the condition of unknown transition probability is one of the problems to be solved. This problem can provide theoretical support for the application of Nash differential game theory in Markov jump systems with partial unknown transition probability to management problems. Based on it, the case of one-player game, which is called the $\epsilon -$suboptimal control problem, is firstly studied. By using the free-connection weighting matrix and “complete square” method, the sufficient conditions for the existence of the $\epsilon -$suboptimal cntrol strategy are obtained, and an explicit expression of the upper bound of the cost function is given. Then, the conditions for the existence of $\epsilon -$suboptimal Nash equilibrium strategy are equivalent to solving the optimization problem, which satisfied the bilinear matrix inequalities (BMIs) and matrix inequalities. The heuristic algorithm is used to solve the optimization problem to obtain the $\epsilon -$suboptimal Nash equilibrium strategies. Finally, the numerical examples are provided to demonstrate the validity of the main conclusions.