ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

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Issue		ESAIM: COCV Volume 5, 2000
Page(s)		369 - 393
DOI		10.1051/cocv:2000114

DOI: 10.1051/cocv:2000114

ESAIM: COCV, July 2000, Vol. 5, 369-393

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

Gianni Dal Maso
SISSA, via Beirut 2, 34014 Trieste, Italy.

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

Received July 26, 1999. Revised April 28, 2000

Abstract: We investigate the value function of the Bolza problem of the Calculus of Variations $\begin{equation*}% V (t,x)=\inf \left\{ \int_ L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^ (0,t;\mathbb{R}^n),\; y(0)=x \right\}, \end{equation*}$
with a lower semicontinuous Lagrangian L and a final cost $\varphi$ , and show that it is locally Lipschitz for t>0whenever 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 and phrases: Discontinuous Lagrangians, Hamilton-Jacobi equations, viability theory, viscosity solutions.

AMS Subject Classification: 49L20, 49L25

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