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Value functions for Bolza problems with discontinuous LagrangiansandHamilton-Jacobi inequalities

Published online by Cambridge University Press:  15 August 2002

Gianni Dal Maso  and
Hélène Frankowska
Gianni Dal Maso
Affiliation:
SISSA, via Beirut 2, 34014 Trieste, Italy.
Hélène Frankowska
Affiliation:
CNRS, ERS2064, Centre de Recherche Viabilité, Jeux, Contrôle, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; [email protected].
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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.

Keywords

Discontinuous LagrangiansHamilton-Jacobi equationsviability theoryviscosity solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 369 - 393
Copyright
© EDP Sciences, SMAI, 2000

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