Hamilton-Jacobi-Bellman equation based on fractional random impulses system

In this talk, we consider the optimal control problem for fractional order random impulses differential equations (1<beta<2), and there is no corresponding Hamilton-Jacobi-Bellman (HJB) equation proposed for fractional order differential equations in existing literature. The lack of semi group properties in fractional order makes the original dynamic programming methods unable to directly handle fractional order problems. We have dealt with this problem by combining the properties of fractional order integrals. In solving this problem, we found that the order of the original system in HJB equation is at least 1. When the order is less than 1, our approach to fractional order ideas provides a possibility to solve the problem.