Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities
Gianni Dal Maso, Hélène Frankowska · ESAIM Control Optimisation and Calculus of Variations · 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.