Hélène Frankowska

Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations

The value function of Mayer’s problem arising in optimal control is investigated, and lower semicontinuous solutions of the associated Hamilton–Jacobi–Bellman equation are defined in three (equivalent) ways. Under quite weak assumptions about the control system, the value function is the unique solution. Moreover, it is stable with respect to perturbations of the control system and the cost. It coincides with the viscosity solution whenever it is continuous.

Existence of neighbouring feasible trajectories: applications to dynamic programming for state constrained optimal control problems

In this paper, the value function for an optimal control problem with endpoint and state constraints is characterized as the unique lower semicontinuous generalized solution of the Hamilton-Jacobi equation. This is achieved under a constraint qualification (CQ) concerning the interaction of the state and dynamic constraints. The novelty of the results reported here is partly the nature of (CQ) and partly the proof techniques employed, which are based on new estimates of the distance of the set of state trajectories satisfying a state constraint from a given trajectory which violates the constraint.

A Connection Between the Maximum Principle and Dynamic Programming for Constrained Control Problems

We consider the Mayer optimal control problem with dynamics given by a nonconvex differential inclusion, whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function. This additional relation is applied in turn to show that the maximum principle is nondegenerate. We also provide a sufficient condition for the normality of the maximum\break principle. To derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories. Then duality theory of convex analysis is applied to derive necessary conditions for optimality. In this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case.

Hölder metric regularity of set-valued maps

This paper is devoted to metric regularity of set-valued maps from a complete metric space to a Banach space. In particular we extend a known characterization of the regularity modulus to maps defined on reflexive spaces. The higher order metric regularity, i.e. an extension of metric regularity to Hölder context, is also investigated using high order variations of set-valued maps and results of similar nature are obtained for conical metric regularity.

Lyusternik-Graves theorem and fixed points

For set-valued mappings and acting in metric spaces, we present local and global versions of the following general paradigm which has roots in the Lyusternik-Graves theorem and the contraction principle: if is metrically regular with constant and is Aubin (Lipschitz) continuous with constant such that , then the distance from to the set of fixed points of is bounded by times the infimum distance between and . From this result we derive known Lyusternik-Graves theorems, a recent theorem by Arutyunov, as well as some fixed point theorems.

Value function and optimality condition for semilinear control problems. II: Parabolic case

In this paper we continue to study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories.

Pointwise Second-Order Necessary Optimality Conditions for the Mayer Problem with Control Constraints

This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem when the control set

The connection between the maximum principle and the value function for optimal control problems under state constraints

U

Conditional visuo-motor learning and dimension reduction

is a closed subset of

Relaxation of Control Systems Under State Constraints

\mathbb{R}^m