Optimal Control Problem for Stochastic Delayed Switching Systems

The paper is concerned with optimization problem of controlled stochastic switching systems. A switching system are regulated by the set of stochastic differential equations which initial circumstances to rely on its antecedent behavior and by the limitations on the transitions or switches between operating mode. Problems of optimization for switching systems play important role in exploretion of modeling nature appearance with the continuous behavior of changing law of dynamics.In general necessary and sufficients conditions of optimality are a significant implement for solution of optimal control problems. It is known that the first-order necessary conditions provide a fundamental tools to investigate optimal control systems. But sometimes the first-order necessary condition cannot give fairly full information to find the acceptable evidence for optimal solutions. Consequently, along such singular controls additional tests of optimality are required. In this research for delayed stochastic switching systems optimality condition on singular controls is established.