Séance Séminaire

Séminaire ACSIOM

Normality of the maximum principle and regularity of minimizers for state constrained optimal control problems

Let us consider a state constrained Bolza optimal control problem. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate. In the first part of the talk we propose sufficient conditions in order to guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be non smooth. In the second part of the talk, we start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy H\"older type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are H\"older continuous, while they have the two sided lower H\"older continuity property for less regular constraints. Finally, we provide sufficient conditions for H\"older type regularity of optimal controls.