| Issue |
ESAIM: COCV
Volume 5, 2000
|
|
|---|---|---|
| Page(s) | 369 - 393 | |
| DOI | https://doi.org/10.1051/cocv:2000114 | |
| Published online | 15 August 2002 | |
Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities
1
SISSA, via Beirut 2, 34014 Trieste, Italy.
2
CNRS, ERS2064, Centre de Recherche
Viabilité, Jeux, Contrôle, Université de
Paris-Dauphine, 75775 Paris Cedex 16, France; frankows@viab.dauphine.fr.
Received:
26
July
1999
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
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