Optimal control of Goursat-Darboux systems with discontinuous co-state

We study an optimal control problem for Goursat-Darboux systems over domains that are more general than rectangles, with costs that include integral terms supported in the two-dimensional domain as well as terms with singular one-dimensional and zero-dimensional support. We prove that the co-state satisfies a system of Hamiltonian equations and a set of side conditions that involves jump discontinuities across a network of characteristic curves. We present a method of solution of the Hamiltonian equations, and in the process of doing so we establish the unique solvability of these equations.