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 $$V(t,x)=\inf \{{\int }_0^{t}L(y(s),y^{\prime }(s))ds+\varphi (y(t)):y\in W^{1,1}(0,t;R^n);y(0)=x\},$$ with a lower semicontinuous Lagrangian $L$ and a final cost $\varphi$, 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.

33 pages. Control, Optimization and Calculus of Variations, to appear