Based on discrete martingale theory, the LaSalle-type theorem for general discrete-time stochastic systems is obtained and the almost sure stability is in turn discussed. As applications, infinite horizon nonlinear optimal regulator is investigated, and a dynamical programming equation called the Hamilton-Jacobi-Bellman equation is also derived for discrete-time nonlinear stochastic optimal control.
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Zhang et al. (2016) studied this question.
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