An optimal control problem for quasilinear differential equations with Bitsadze–Samarski boundary conditions

The present paper is devoted to optimal control problems whose behavior is described by quasilinear first-order differential equations on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem on the existence and uniqueness of a generalized solution in the space ${C}_{\mu}\left(\overline{G}\right)$ is proved for quasilinear differential equations; necessary optimality conditions are obtained in terms of the maximum principle; the Bitsadze–Samarski boundary-value problem is examined for a first-order linear differential equation; the existence of a solution in the space ${C}_{\mu}^p\left(\overline{G}\right)$ is proved, and an a priori estimate is derived. A necessary and sufficient optimality condition is proved for a linear optimal control problem.