A verification theorem for hybrid optimal control problem

A class of the hybrid optimal control problem is formulated and a hybrid dynamic programming principle (DPP) is presented which constitutes a generalization of the celebrated dynamic programming principle of Richard Bellman. It is shown that similarly to the case of continuous dynamic programming, which leads to the well known Hamilton-Jacobi-Bellman (HJB) functional partial differential equation, the hybrid dynamic programming principle (HDPP) leads to a hybrid HJB equation.