Deterministic Exit Time Control Problems With Discontinuous Exit costs

We study deterministic exit time control problems with discontinuous exit costs. When the exit cost $\phi$ is upper semicontinuous and there is an outer field on the boundary, we show that all the value functions have the same lower semicontinuous envelope which is the unique lower semicontinuous viscosity solution of the associated Dirichlet problem. We also prove uniqueness results for the generalized Dirichlet problem for first-order Hamilton--Jacobi equations with convex Hamiltonians and with discontinuous boundary conditions, under some nondegeneracy conditions on the Hamiltonians on the boundary.