A new approach for second-order linear matrix descriptor differential equations of Apostol-Kolodner type (Q2870739)

This paper is concerned with the solution of the second-order linear matrix differential equation NEWLINE\[NEWLINE FX''(t)=GX'(t)+AX(t), \tag{1} NEWLINE\]NEWLINE where \(F,\,G,\,A\) are real (or complex) square matrices of size \(n\) by \(n\), \(F\) is singular, i.e., \(\det F=0\), and \(X\) is an unknown rectangular matrix-valued function of size \(n\) by \(m\). The equation (1) is called a descriptor differential equation of Apostol-Kolodner type. The main aim of the paper is to find an explicit formula for the solution of (1) that satisfies the given consistent initial conditions \(X(0)=\mathbb O\) and \(X'(0)\neq \mathbb O\). By reducing (1) to a first-order system and using the Weierstrass canonical form of the pencil \(sF-G\), an analytical formula for the solution is obtained.