Optimal control of Goursat-Darboux systems with discontinuous co-state (Q878919)

The author studies an optimal control problem for a system governed by the Goursat-Darboux PDE of the following form \[ \frac{\partial^2x(s,t)}{\partial s \, \partial t} = f(s,t,x(s,t),x_s(s,t),x_t(s,t),u(s,t)) \] over a domain more general than a rectangle, in particular, the part of its boundary is supposed to be a piece-wise smooth curve. The author derives the Hamiltonian equations for the co-state via its explicit representation in terms of the matrix-valued Riemann function and describes side conditions for these equations involving the jumps of the co-state across a network of characteristic curves. The unique solvability of the Hamiltonian equations is proved.