Near-optimal Control Problem Of Stochastic Systems

In this paper, we have studied the control problem of stochastic population system constructed by the stochastic differential equations. As known to all, for the optimal control it needs strict conditions relatively, and also not any one has its own optimal control. Therefore, some people considered whether can we get a near-optimal control of the problem with lower conditions. So in our paper, we mainly study the near-optimal control. The main research in the following areas:1. With the adjoint equations and Hamiltonian functions, using Ito’s formula, Holder inequality, Barkholder-Davis-Gundy’s inequality, we study the near-optimal control of stochastic age-structured pop-ulation systems, and also give the necessary and sufficient conditions of the near-optimal control. At last, an example is given for illustration.2. Here, we investigate near-optimality of stochastic predator-prey population systems of two species. Using Ito’s formula, Ekeland’s principle and maximum principle, and so on, we also give out the necessary and sufficient conditions of near-optimal control of the systems.3. For better description of the population systems, we introduce the Markov switching and Poisson jumps into the systems, and discuss the near-optimal control problem of stochastic predator-prey popu-lation systems of two species with Markov switching and Poisson jumps. At last, using Ito’s formula, Holder inequality, Ekeland’s principle and maximum principle, we give out the necessary and sufficient conditions of near-optimal control of the systems.