Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities
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2 CNRS, ERS2064, Centre de Recherche Viabilité, Jeux, Contrôle, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; email@example.com.
Revised: 28 April 2000
We investigate the value function of the Bolza problem of the Calculus of Variations with a lower semicontinuous Lagrangian L and a final cost , 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.
Mathematics Subject Classification: 49L20 / 49L25
Key words: Discontinuous Lagrangians / Hamilton-Jacobi equations / viability theory / viscosity solutions.
© EDP Sciences, SMAI, 2000