SUMMARY
The article describes the boundary-value problem of the theory of optimal control for matrix differential equations in the critical case. This problem is urgent for wide range of applications like modeling of linear technological and economic processes. A problem of controlling movement is reduced to solutions of boundary-value problems of n× n systems of ordinary differential equations of the first order. Solution of these tasks is very complex in case of control of systems in which the number of processes exceeds the amount of information about the initial state. The paper is devoted to finding conditions for solvability and construction solutions of boundary-value problems in critical case of the theory of motion control, in which the number of boundary conditions does not coincide with the number of unknowns in the system. The condition of solvability of such problems is found. Author describes the approach for solution to the problem with the help of theory of pseudoinverse matrices. In addition this approach is applicable to the problem of analytic construction of regulators and the optimal stabilization.