Solutions of generalized linear matrix differential equations which satisfy boundary conditions at two points

Abstract

In this article, we study a boundary value problem of a class of generalized linear matrix differential equations whose coefficients are square constant matrices. By using matrix pencil theory we obtain formulas for the solutions and we give necessary and sufficient conditions for existence and uniqueness of solutions. Moreover we provide some numerical examples. These kinds of systems are inherent in many physical and engineering phenomena.

The Samuelson's model as a singular discrete time system

“…The class of sF − G is characterized by a uniquely defined element, known as the Weierstrass canonical form, see [50][51][52][53][54][55][56][57], specified by the complete set of invariants of sF − G. This is the set of elementary divisors of type (s − a j ) pj , called finite elementary divisors, where a j is a finite eigenvalue of algebraic multiplicity p j (1 ≤ j ≤ ν), and the set of elementary divisors of type ŝq = 1 s q , called infinite elementary divisors, where q is the algebraic multiplicity of the infinite eigenvalue. ν j=1 p j = p and p + q = m.…”

Causality of singular linear discrete time systems

“…Throughout the paper we will use in several parts matrix pencil theory to establish our results. A matrix pencil is a family of matrices sF −G, parametrized by a complex number s, see [46][47][48][49][50][51][52][53].…”

“…In this article we consider the system (1) with a regular pencil, where the class of sF − G is characterized by a uniquely defined element, known as the Weierstrass canonical form, see [46][47][48][49][50][51][52][53], specified by the complete set of invariants of sF − G. This is the set of elementary divisors of type (s − a j ) pj , called finite elementary divisors, where a j is a finite eigenvalue of algebraic multiplicity p j (1 ≤ j ≤ ν), and the set of elementary divisors of type ŝq = 1 s q , called infinite elementary divisors, where q is the algebraic multiplicity of the infinite eigenvalue. ν j=1 p j = p and p + q = m.…”

Homogeneous linear matrix difference equations of higher order: Singular case

“…Let d=dimN r (sF − G) and t=N l (sF − G). It is known [46][47][48][49][50][51][52][53] that N r (sF − G), N l (sF −G), as rational vector spaces, are spanned by minimal polynomial bases of minimal degrees…”

“…respectively. The set of minimal indices ǫ i and ζ j are known [46][47][48][49][50][51][52][53] as column minimal indices (c.m.i.) and row minimal indices (r.m.i) of sF-G respectively.…”