NECESSARY AND SUFFICIENT CONDITIONS OF OPTIMALITY FOR HYPERBOLIC EQUATION WITH NON-LOCAL BOUNDARY CONDITIONS

The theory of boundary value problems for hyperbolic equations is one of the most important sections of the theory of partial differential equa-tions, which is explained both by the theoretical significance of the results and by the presence of their practical applications in gas dynamics, in the theory of infinitesimal bendings of surfaces, in the momentless theory of shells, in magnetic hydrodynamics, mathematical biology and other areas. In the XX century in connection with applied technical and economic needs, it became necessary to solve control and optimization problems. The most fa-mous works in this area are J.I. S. Pontryagin and his students V. G. Boltyan-skiy, R. V. Gamkrelidze, E. F. Mishchenko, studied issues of control of pro-cesses described by systems of ordinary differential equations [18,19]. Further development of applied research leads to the need to control ob-jects, the behavior of which is described using methods with partial differential equations. The corresponding control problems were considered in the works of A.G Butkovsky and co-authors [3,4], A.I Egorov [6-9], J.-L. Lyons [12], K. A. Lurie [13], T. K. Sirazetdinov [20], as well as V. A. Ilyin and E. I. Moiseeva [21,22], [14,15], A.T.Ramazanova [26], Kuliyev Hamlet [27]. Optimal control problems for the processes described by hyperbolic differ-ential equations are encountered in various applications [1,2]. Such problems have been most fully investigated in cases where the boundary conditions for the equations of state are classical. However, there are numerous problems in physics, technology, biology, etc., in which the processes are described by hyperbolic equations, where the boundary conditions are nonclassical or non -local [23,24]. Among the nonlocal boundary value problems for the hyperbolic equations , a special place is occupied by boundary value problems with inte-gral boundary conditions [25]. Optimal control problems for hyperbolic-type equations with nonlocal boundary conditions, including those with integral boundary conditions, have been little studied.