An Asymptotic Method for Predicting Risks in Problems of Stochastic Monitoring and Control

Abstract

To ensure the stabilization of the equilibrium state in a nonlinear system in the presence of noise, it is not enough to solve the local stabilization problem, it is also necessary to ensure continuous monitoring of a possible transition to a critical state leading to system failure. To organize such monitoring, we use the large deviations principle applied to dynamical systems with small perturbations. For the purposes of monitoring, the optimal path that we call the A-profile is important. We use the A-profile to build a situational forecast in the risk control problem for a multi-agent system. In addition to the nonlinear mechanism of internal stabilization of the level h for each of the agents, there are forces of mean field interaction between the agents. The weak limit in this model with the number of agents tending to infinity is described by the FokerPlanck-Kolmogorov equation, but the use of approximation up to O(h2) leads to a finite-dimensional Wentzel-Freidlin scheme. According to the scheme, we obtain an explicit A-profile as a solution of the degenerate Abel equation of the second kind. At the same time, the approximation in h makes it possible to develop a method of successive approximations for the A-profile. In this paper, the A-profile is synthesized as a solution of the optimal control problem, where the state-dependent Riccati equation method and the method of the approximating sequence of Riccati equations are used. In the article, these methods are applied and compared within the framework of the risk control problem.