Domain Recurrence and Probabilistic Analysis of Residence Time of Stochastic Systems and Domain Aiming Control
The problem of domain aiming control is formulated for controlled stochastic nonlinear systems. This issue involves regularity of the solution to the resulting closed-loop stochastic system. To begin with, an extended existence and uniqueness theorem for stochastic differential equation with local Lipschitz coefficients is proven by using a Lyapunov-type function. A Lyapunov-based sufficient condition is also given under which there is no regularity of the solution for a class of stochastic differential equations. The notions of domain recurrence and residence time for stochastic nonlinear systems are introduced, and various criteria for the recurrence and non-recurrence relative to a bounded open domain or an unbounded domain are provided. Furthermore, upper bounds of either the expectation or the moment-generating function of the residence time are derived. In particular, a connection between the mean residence time and a Dirichlet problem is investigated and illustrated with a numerical example. Finally, the problem of domain aiming control is considered for certain types of nonlinear and linear stochastic systems. Several examples are provided to illustrate the theoretical results.