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Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities. (English) Zbl 0952.49024

Summary: We investigate the value function of the Bolza problem of the Calculus of Variations \[ V(t,x)=\inf\left\{\int^t_0L(y(s),\;y^\prime(s))ds+\varphi(y(t)):y\in W^{1,1} (0,t;\mathbb{R}^n),\;y(0)= x\right\}, \] 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.

MSC:

49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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