Quasi-singular control in Goursat- Darboux stochastic systems

In a stochastic control problem described by a nonlinear hyperbolic equation with a Goursat-Darboux type boundary condition, first-order necessary optimality condition in the form of linearized Pontryagin maximum principle is obtained. The case of degeneration of the linearized maximum principle is considered, and various necessary optimality conditions are established for quasi-singular (i.e., controls for which the linearized maximum condition degenerates) controls.