Hamilton-Jacobi conditions for a class of impulsive control problems

Abstract

This work presents optimality conditions ofHamilton-Jacobi type for a classe of vector-valuedimpulsive control optimal problems. The dynamicsare defined by a measure driven differentialinclusion and the vector fields associated with thesingular term do not satisfy the so called Frobeniuscondition. The concept of verification function forthe class of problems addressed here is presented.Besides some regularity hypotheses, verificationsfunctions satisfy a set of Hamilton-Jacobi typeconditions, as well as a given boundary condition.It is shown that the existence of a verificationfunction is a necessary and sufficient condition forthe optimality of a feasible trajectory (in the senseof proper solution). It is also shown that the valuefunction of the family of problems parametrized bythe initial date is a verification function, with someextra properties, and results relating subgradientsof the value function and multipliers of necessaryconditions of the Maximum Principle are presented,too.