Abstract
This paper is devoted to the optimal control problem of switching system in which constraints on the state variable are given by inclusions. Using Ekeland’s variational principle, second-order necessary condition of optimality for stochastic switching systems with constraints is obtained, and transversality conditions for switching law are established.
Highlights
1 Introduction Stochastic differential equations have provided a lot of interest for problems of nuclear fission, communication systems, self-oscillating systems, etc., where the influences of random disturbances cannot be ignored [, ]
The stochastic maximum principle has been first considered by Kushner [ ]
A general theory of the stochastic maximum principle based on random convex analysis was given by Bismut [ ]
Summary
Stochastic differential equations have provided a lot of interest for problems of nuclear fission, communication systems, self-oscillating systems, etc., where the influences of random disturbances cannot be ignored [ , ]. Modern presentations of stochastic maximum principle with backward stochastic differential equations are considered in [ – ]. First-order necessary conditions of optimality for stochastic switching systems have been studied by the author in [ – ]. Deterministic optimization problems for singular controls were intensively used by [ , ] Such kind problems for stochastic systems have been investigated in [ , ]. The singular optimal control problem of stochastic switching systems with uncontrolled diffusion coefficients is considered. Second-order necessary conditions of optimality and transversality conditions are obtained in [ ] for uncontrolled switching systems. Backward stochastic differential equations and Ekeland’s variational principle is used to establish a singular maximum principle for stochastic optimal control problems of constrained switching systems
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