A Maximum Principle for State-Constrained Optimal Sweeping Control Problems

This letter concerns a Maximum Principle for a state-constrained optimal control problem whose dynamics is given by a sweeping process. The main novelty for this class of problems is the additional ingredient, a state constraint, in the context of sweeping processes. The derived optimality conditions are in the Gamkrelidze’s form which has the virtue of ensuring a much greater regularity of the multiplier associated with the state constraint. In fact, this multiplier is a monotonic function of bounded variation instead of the positive measure that appears in the Maximum Principle in the Dubovistkii-Milyutin form. The relation between these two forms of necessary conditions of optimality is established. Another novel aspect in the context of sweeping processes concerns the consideration of a solution concept for measure driven differential equations to be satisfied by the adjoint variable. An additional controllability assumption is considered in order to ensure the nondegeneracy of the Maximum Principle conditions.