Solution Properties of Linear Descriptor (Singular) Matrix Differential Systems of Higher Order with (Non-) Consistent Initial Conditions

Athanasios A Pantelous,Grigoris I Kalogeropoulos,Kostas G Arvanitis,Athanasios D Karageorgos

doi:10.1155/2010/897301

Abstract

In some interesting applications in control and system theory, linear descriptor (singular) matrix differential equations of higher order with time-invariant coefficients and (non-) consistent initial conditions have been used. In this paper, we provide a study for the solution properties of a more general class of the Apostol-Kolodner-type equations with consistent and nonconsistent initial conditions.

Highlights

IntroductionLinear Time-Invariant LTI i.e., with constant matrix coefficients descriptor matrix differential systems of type 1.1 with several kinds of inputs
Linear Time-Invariant LTI i.e., with constant matrix coefficients descriptor matrix differential systems of type 1.1 with several kinds of inputsFX r t AX t BU t, 1.1 where F, A ∈ M n × m; F, B ∈ M n × μ; F, and U ∈ C∞ F, M μ × m; F, often appear in control and system theory
We study the class of LTI descriptor singular matrix descriptor differential equations of higher order whose coefficients are rectangular constant matrices

Summary

Introduction

Linear Time-Invariant LTI i.e., with constant matrix coefficients descriptor matrix differential systems of type 1.1 with several kinds of inputs. System 1.1 might be considered as the more general class of higher order linear descriptor matrix differential equations of Apostol-Kolodner type, since Kolodner has studied such systems in nondescriptor form; see 8. Adopting several different methods for computing the matrix powers and exponential, new formulas representing auxiliary results are obtained This allows us to prove properties of a large class of linear matrix differential equations of higher order; in particular results of Apostol and Kolodner are recovered; see 5, 8. It should be mentioned that in the classical theory of linear descriptor differential systems, see, for instance, 1, 2, 11–13 , one of the important features is that not every initial condition X0 admits a functional solution. It is not rare to appear in some practical significant applications that the assumption of the initial conditions for 1.3 can be nonconsistent, that is, X to / X0

Mathematical Background and Notations

Solution Space for Consistent Initial Conditions

Solution Space Form of Nonconsistent Initial Conditions

Conclusions