with a lower semicontinuous Lagrangian L and a final cost ,andshow that it is locally Lipschitz for t>0whenever L is locally bounded. It also satisfiesHamilton-Jacobi inequalities in a generalized sense.When the Lagrangian is continuous, then the value function is theunique lower semicontinuous solutionto the corresponding Hamilton-Jacobi equation, while for discontinuousLagrangian we characterize the value function by using the socalled contingent inequalities." name="description" /> Cambridge Journals Online - ESAIM: Control, Optimisation and Calculus of Variations - References - Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2000 - Volume 5

Research Article

Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

Maso, Gianni Dala1 and Frankowska, Hélènea2

a1 SISSA, via Beirut 2, 34014 Trieste, Italy.

a2 CNRS, ERS2064, Centre de Recherche Viabilité, Jeux, Contrôle, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; frankows@viab.dauphine.fr.

Abstract

We investigate the value function of the Bolza problem of the Calculus of Variations
 $$ V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\},$$ with a lower semicontinuous Lagrangian L and a final cost $ \varphi $ , and show that it is locally Lipschitz for t>0 whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities.

(Received July 26 1999)

(Revised April 28 2000)

(Online publication August 15 2002)

Key Words:

  • Discontinuous Lagrangians;
  • Hamilton-Jacobi equations;
  • viability theory;
  • viscosity solutions.

Mathematics Subject Classification:

  • 49L20;
  • 49L25
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