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The problem of domain aiming control is formulated for controlled stochasticnonlinear systems. This issue involves regularity of the solution to theresulting closed-loop stochastic system. To begin with, an extended existenceand uniqueness theorem for stochastic differential equation with localLipschitz coefficients is proven by using a Lyapunov-type function. ALyapunov-based sufficient condition is also given under which there is noregularity of the solution for a class of stochastic differential equations.The notions of domain recurrence and residence time for stochastic nonlinearsystems are introduced, and various criteria for the recurrence andnon-recurrence relative to a bounded open domain or an unbounded domain areprovided. Furthermore, upper bounds of either the expectation or themoment-generating function of the residence time are derived. In particular, aconnection between the mean residence time and a Dirichlet problem isinvestigated and illustrated with a numerical example. Finally, the problem ofdomain aiming control is considered for certain types of nonlinear and linearstochastic systems. Several examples are provided to illustrate the theoreticalresults.

Domain Recurrence and Probabilistic Analysis of Residence Time of Stochastic Systems and Domain Aiming Control