Journal of Mathematical Sciences, Vol. 206, No. 4, April, 2015 AN OPTIMAL CONTROL PROBLEM FOR QUASILINEAR DIFFERENTIAL EQUATIONS WITH BITSADZE–SAMARSKI BOUNDARY CONDITIONS D. Devadze and V. Beridze UDC 517.9 Abstract. 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 Cμ( G) 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 p μ ( G) is proved, and an a priori estimate is derived. A necessary and sufficient optimality condition is proved for a linear optimal control problem. CONTENTS 1. Existence of a generalized solution ............................. 358 2. Linear problem ....................................... 361 3. Statement of the optimal control problem ........................ 362 4. Construction of the adjoint equation in the differential form .............. 365 5. Necessary and sufficient conditions of optimality ..................... 366 References ............................................. 369 For many optimization problems in elasticity theory, mechanics, diffusion processes, kinetics of chemical reactions, etc., the state of a system is described by partial differential equations (see [2, 7, 12, 18]). Therefore, control problems for systems with distributed parameters attract great attention (see [2, 3, 5]). The nonlocal Bitsadze–Samarski boundary-value problem (see [1]) arose in connection with the mathematical modeling of processes occurring in plasma physics. The present paper is devoted to optimal control problems whose behavior is described by quasi- linear first-order differential equation on the plane with nonlocal Bitsadze–Samarski boundary con- ditions. When dealing with optimization problems for systems with distributed parameters, it is important to investigate questions of the existence of generalized solutions for discontinuous right- hand sides of an equation (see [11, 17, 19]). In this paper, the necessary conditions of optimality are obtained by using the approach elaborated in [15, 16] for control systems of the general form. Nonlocal Bitsadze–Samarski boundary-value problems are studied by means of an algorithm that reduces non- local boundary-value problems to a sequence of Riemann–Hilbert problems for elliptic equations. This method makes it possible not only to solve the problem numerically, but also to prove the existence of a solution (see [8, 9]). In this paper, the theorem on the existence and uniqueness theorem of a generalized solution in the space C μ ( G) is proved for quasilinear first-order differential equations with nonlocal boundary condi- tions; the Bitsadze–Samarski boundary-value problem is considered for a first-order linear differential Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 89, Differential Equations and Mathematical Physics, 2013. 1072–3374/15/2064–0357 c  2015 Springer Science+Business Media New York 357 DOI 10.1007/s10958-015-2317-5