Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data

Julien Bernis,Piernicola Bettiol

doi:10.1051/cocv/2019041

Abstract

We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.

Highlights

Consider the non autonomous Bolza problem: Minimize T SL(t, x(t), x (t))dt + g(x(T )) (PS,x0 )over x (t) arcs x ∈ W 1,1([S, T ], Rn) satisfying ∈ F (t, x(t)) for almost every t ∈ [S, T], x(S) = x0, in which [S, T ] is a given interval, x0 ∈ Rn is a given initial datum, g : Rn → R ∪ {+∞} and L : [S, T ] × Rn × Rn → R are given functions, and F : [S, T ] × Rn Rn is a given multivalued function
As a consequence the natural class of functions in which we study the value function is the set of lower semicontinuous functions
We introduce the auxiliary Lagrangian L− which will be used as a technical tool in the characterization of solutions to the Hamilton-Jacobi equation

Summary

Introduction

Over x (t) arcs x ∈ W 1,1([S, T ], Rn) satisfying ∈ F (t, x(t)) for almost every t ∈ [S, T. A different perspective has been recently suggested in [4] for the intermediate case (between the continuous one and the merely measurable one) when the multifunction t F (t, x) has everywhere one-sided limits, for all x, and is continuous on the complement of a zero-measure subset of [S, T ] (without necessarily imposing further a priori regularity conditions such as the absolute continuity of the epigraph of the candidate solutions) In this context, considering optimal control problems with a final cost term (i.e. L = 0), the value function turns out to be the unique lower semicontinuous solution to (HJE) taking into account ‘everywhere in t’ characterizations which involve the concepts of lower Dini derivative and the proximal subdifferential.

Notation

Hypotheses

A priori boundedness and hypotheses reduction technique

Preliminary results

The value function is a proximal solution

A proximal solution coincides with the value function: comparison results