On Symmetric Solutions of Linear Time-Varying Matrix Differential Equations
Linear matrix differential equations are of great interest to many branches of science and technology. For instance, an analysis of solutions to linear time-varying matrix differential equations may be required when solving terminal control problems for nonlinear control systems. It has been proven recently that a solution to the initial value problem for a linear matrix differential equation with analytical coefficients is symmetric in a simply connected domain of the complex plane if and only if all the derivatives of the solution calculated along trajectories of the equation are symmetric at the initial point. In the paper, we address the problem of solutions symmetry for linear time-varying matrix differential equations with coefficients of a finite degree of smoothness. First of all, we prove a sufficient condition for the solution symmetry on a given interval. To check whether or not the solution of an initial value problem is symmetric on an interval, one should construct two special matrix sequences and calculate successive derivatives of the solution along trajectories of the equation. If all the derivatives up to some order are symmetric at the initial point and a given set of properties is met for the matrix sequences, then the solution is symmetric over the whole interval. Provided the proposed condition is met, we establish a formula for symmetric solutions of the equation. We demonstrate how this formula enables us to find a symmetric solution to the equation in the case when direct solving does not seem possible for the original equation. Finally, we show that the obtained formula can be applied to construct estimates for symmetric solutions of the equation. This is especially meaningful in the case when the direct use of the formula does not simplify appropriate calculations to find solutions to the original equation.Further research in this field should be aimed at obtaining novel estimates for the spectrum bounds of symmetric solutions to linear matrix time-varying differential equations. The results of the present work may be interesting for those who deal with constructing solutions to terminal problems for nonlinear control systems.