Studying problems with nonlocal conditions for differential equations is stimulated by various circumstances, in particular solving problems in the theory of plasma physics, inverse problems for parabolic equations. The popularity of research in control systems, described by differential equations with partial derivatives, is associated with their use in solving the problems of natural science, in particular, hydro and gas dynamics, heat physics, filtration, diffusion, plasma, and the theory of biological populations.

In this paper, the problem of optimal control of a system described by the oblique derivative problem and the integral condition for a time variable for a second order parabolic equation with power singularities in the equation and boundary condition coefficients is investigated. The cases of internal, starting and border control are considered. The quality criterion is given by the sum of bulk and surface integrals.

With the help of modified methods developed in the study of boundary value problems for parabolic equations with smooth coefficients, a priori estimates, the existence and uniqueness of the solution of a nonlocal parabolic boundary-value problem with degeneracy was established. The coefficients of the parabolic equation and the boundary condition admit the power singularities of an arbitrary order of any variables on a certain set of points. The estimates of the solution of a nonlocal boundary value problem and its derivatives in Hölder space with a power weight are determined, which is determined by the order of degeneration of the coefficients of the equation and the boundary condition.

Using the integral image of solutions of the sequence of auxiliary nonlocal boundary value problems with smooth coefficients, the problem of optimal control of a system described by a nonlocal parabolic problem is investigated. Using the Arzel theorem and the methods of the variational calculus, necessary and sufficient conditions for the existence of an optimal solution of a system described by a nonlocal boundary-value problem with an integral condition for a time variable for parabolic equations with degenerate coefficients are establish. The values of optimal internal, starting and boundary control and the estimation of the optimal solution are found. 

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