Dziuba M. Differential-algebraic matrix boundary value problems.

The thesis research is devoted to the study of the problem of finding constructive conditions for the existence and construction of solutions of matrix differential-algebraic boundary value problems in the case when an unknown is a matrix function. The matrix record of an unknown generalizes the form of a matrix differential-algebraic equation, as well as a boundary condition. In the study of differential algebraic boundary value problems, the following fact is a significant obstacle for the use of traditional methods of studying periodic and Noetherian boundary value problems: even the Cauchy problem for differential algebraic systems, which was investigated by S.Campbell, A.M.Samoilenko, M.O.Perestyuk, Yu.E.Boyarintsev, V.F.Chistyakov and O.A.Boichuk, in general, is not solvable for arbitrary initial values. In the dissertation, with the help of the tool of pseudo-inverse matrices, the scheme of study of the problem of the existence and construction of solutions of matrix differential-algebraic boundary value problems was improved. An example of the Lyapunov, Sylvester and Riccati matrix equations demonstrates the efficiency of the solvability conditions and the solutions for the construction of solutions. The scheme of regularization of the Lyapunov and Sylvester matrix equations is constructed, which differs significantly from the classical Tikhonov regularization method. On the example of matrix periodic and multipoint problems for differential algebraic equations, the efficiency of the obtained solvability conditions and the scheme of construction of solutions are demonstrated. Keywords: differential-algebraic boundary value problems, matrix equations, differential-algebraic equations, pseudo-inverse matrices, generalized Green operator.