A reliability system analysis with stochastic differential equation and non-linear DP equation

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1 Purposes of this study are (1) When coefficients of a linear system obey a markov chain, I obtain the stability conditions of the probability linear system. (2) When the probability linear system is stable, I obtain relations with the optimal control policy and viscosity solution of a non-linear DP equation under a suitable criterion. (3) I study algorithms to obtain the solution of a stochastic differential equation.2 Last year I examined the stability of the linear system with randome parameters obeyed a Markovchain. In particular I studied relations with an exponential function of Martingale and a stochastic differential equation. I adapt this method in analysis of the linear system with random parameters obeyed a Markov chain.This year I studied the Q-learning for Markov decision processes, and presented Research Institute for Mathematical Sciences KOKYUROKU "Mathematical decision making under uncertainty."