Constructing a Solution of a Two-Point Boundary Value Problem for a Nonlinear Matrix Lyapunov Equation

Abstract

We consider the boundary value problem

$$dX/dt=A(t)X+XB(t)+F(t,X), \quad MX(0)+NX(\omega )=0, \quad X\in \mathbb{R}^{n\times n},$$

where $M$ and $N$ are constant $n\times n$ matrices and $A$, $B$, and $F$ are continuous $n\times n$ matrix functions defined on a closed interval $I$ and the direct product $I\times D$, respectively; here $D$ is a ball centered at zero in the space $\mathbb{R}^{n\times n}$. Constructive sufficient conditions for the unique solvability of this problem in terms of the initial data are obtained. An iterative algorithm with an implicit computational scheme is presented. All approximate solutions produced by the algorithm satisfy the given boundary condition.