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

Abstract

We investigate the value function of the Bolza problem of the Calculus of Variations V (t;x )=i nf t 0 L(y(s);y 0 (s))ds +'(y(t)) : y2 W 1;1 (0;t; n ) ;y (0) = x ; with a lower semicontinuous Lagrangian L and a nal cost ', and show that it is locally Lipschitz for t> 0 wheneverL is locally bounded. It also satises 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.