Journal of Mathematical Sciences, Vol. 206, No. 4, April, 2015 METHODS OF NUMERICAL SOLUTION OF OPTIMAL CONTROL PROBLEMS BASED ON THE PONTRYAGIN MAXIMUM PRINCIPLE D. Devadze and V. Beridze UDC 517.9 Abstract. In this paper, we study optimal control problems whose behavior is described by second- order differential equations with nonlocal Bitsadze–Samarski boundary conditions. Necessary condi- tions of optimality are obtained in terms of the maximum principle; adjoint equations are constructed in the differential and integral form. Necessary and sufficient optimality conditions are obtained for a linear problem, a difference scheme is constructed and examined, and a numerical algorithm is pro- posed. CONTENTS 1. Statement of the problem .................................. 348 2. Derivation of necessary optimality conditions ....................... 349 3. Construction of the adjoint equation ........................... 350 4. Necessary and sufficient optimality conditions for a linear problem ........... 352 5. Difference scheme and convergence ............................ 352 6. Computational algorithms ................................. 354 References ............................................. 355 In this paper, for second-order nonlinear ordinary differential equations with Bitsadze–Samarski nonlocal boundary conditions (see [1]), we obtain necessary optimality conditions in the form of the maximum principle (see [11, 12]) and construct adjoint equations in the differential and integral form. Using necessary and sufficient optimality conditions, we reduce the solution of a linear optimal control problem to the solution of an equivalent system of differential equations, whose numerical solution is constructed and the convergence of the difference scheme is examined for the class of functions that have absolutely continuous first derivatives. An algorithm for the numerical solution of a linear optimal control problem is proposed. 1. Statement of the problem. Let −∞ <a<b< +∞, R be the set of real numbers, V be a bounded subset V ⊂ R, and  C 1 [a, b] be the set of functions u :[a, b] → R that are absolutely continuous together with their first derivatives. Each function v :[a, b] → V is called a control. A function v is called an admissible control if v ∈ L 2 [a, b]. The set of all admissible controls is denoted by Ω. For each v ∈ Ω, consider the following Bitsadze–Samarski boundary-value problem: u  = f (t, u, u  ,v), a < t < b, (1.1) u(a)= α, u(b)= u(t 0 ), (1.2) Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 89, Differential Equations and Mathematical Physics, 2013. 348 1072–3374/15/2064–0348 c  2015 Springer Science+Business Media New York DOI 10.1007/s10958-015-2316-6