Sufficient optimality conditions in the form of the Pontryagin maximum principle in control problems for hybrid systems

We generalize the sufficient conditions of the classical optimality theory to a class of optimal control problems for hybrid systems. For the cases of global and strong local extrema we obtain general sufficient optimality conditions and sufficient conditions in the form of the Pontryagin maximum principle. All results rest on dealing with exterior approximations of the attainability sets of controlled systems which are constructed using the solution sets to one of the Hamilton–Jacobi inequalities (strongly monotone functions of Lyapunov type).