On first-order quasi-variational inequalities with integral terms (Q1178310)

The paper deals with first order quasi-variational inequalities with integral terms associated with impulsive and switching control of piecewise-deterministic processes. For these kinds of problems, by considering characteristic solutions in place of viscosity ones an existence and a unicity theorem for the solution are proved, generalizing some results of Capuzzo-Dolcetta, Evans, Lenhart, Li and Blankenship. The proofs are self-contained and based on methods of convex analysis, a discussion on the use of characteristic solutions and of viscosity solutions is also made. Finally some applications of the above theorem to inventory-type impulse cost and to switching control are discussed and a comparison with known results is made.