Optimality conditions and solution algorithms of optimal control problems for nonlocal boundary-value problems

In the present paper, the Bitsadze–Samarski boundary-value problem is considered for a quasi-linear differential equation of first order on the plane and the existence and uniqueness theorem for a generalized solution is proved; the necessary (in the linear case) and sufficient optimality conditions for optimal control problems are found. The optimal control problem is posed, where the behavior of control functions is described by elliptic-type equations with Bitsadze–Samarski nonlocal boundary conditions. The necessary and sufficient optimality conditions are obtained in the form of the Pontryagin maximum principle and the solution existence and uniqueness theorem is proved for the conjugate problem. Nonlocal boundary-value problems and conjugate problems are solved by the algorithm, which reduces nonlocal boundary value problems to a sequence of Dirichlet problems. The numerical method of solution of an optimal control problem by the Mathcad package is presented.