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

G. Dal Maso·H. Frankowska
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Abstract

We investigate the value function of the Bolza problem of the Calculus of Variations $$ V (t,x)=\inf \{\int_{0}^{t} L (y(s),y'(s))ds + ϕ(y(t)) : y \in W^{1,1} (0,t; R^n) ; y(0)=x \}, $$ with a lower semicontinuous Lagrangian $L$ and a final cost $ϕ$, 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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G. Dal Maso, H. Frankowska. 2000-06-02. Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi Inequalities. https://arxiv.org/abs/math/0006013

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