First passage time of stochastic differential systems: Probability estimation and probability guaranteed control

Abstract

This paper is concerned with the probability estimation and probability guaranteed control problems for the first passage time (FPT) of a class of nonlinear systems described by stochastic differential equations (SDEs) within a given boundary. Taking advantage of the comparison theorem of the SDE, a one-dimensional conservatism process is constructed as an upper bound on the process induced by the boundary condition, based on which the FPT probability can be calculated either analytically or numerically. Furthermore, by virtue of the stochastic analysis and the matrix inequality technique, sufficient conditions are established to ensure that such probabilities exceed certain prescribed thresholds in two different scenarios: in the stay-in scenario, the state trajectory is ensured to remain within a given region for a certain period, and in the entry scenario, the state trajectory is ensured to enter a given region within a certain period. Moreover, as an application, the output feedback controller is designed so that the FPT probability of the closed-loop system is guaranteed to exceed a specified index. Finally, numerical examples are presented to validate the effectiveness of the proposed theoretical results.