Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities

Details Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities

2000-06-02

arxiv.org/abs/math/0006013v1

Mathematics

Analysis of PDEs

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

Scientific paper

We investigate the value function of the Bolza problem of the Calculus of Variations $$ V (t,x)=\inf \{\int_{0}^{t} L (y(s),y'(s))ds + \phi(y(t)) : y \in W^{1,1} (0,t; R^n) ; y(0)=x \}, $$ with a lower semicontinuous Lagrangian $L$ and a final cost $\phi$, 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.

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