Abstract
= A(t)X +XB(t)+F(t)where A(t), B(t) and F(t) are continuous n×n matrix-valued functionson R . The basic problem under consideration is the determination ofnecessary and sufficient conditions for the existence of a solution withsome specified boundedness condition. A Clasical result of this type,for system of differential equations is given by Coppel [4, Theorem 2,Chapter V].The problem of Ψ-boundedness of the solutions for systems of ordi-nary differential equations has been studied in many papers, [1, 2, 3, 5,9, 10]. Recently [11, 7], extended the concept of Ψ-boundedness of thesolutions to Lyapunov matrix differential equations. In [6], the authorobtained necessary and sufficient conditions for the non homogenoussystem x
Highlights
The importance of matrix Lyapunov systems, which arise in a number of areas of control engineering problems, dynamical systems, and feedback systems are well known
A Clasical result of this type, for system of differential equations is given by Coppel [4, Theorem 2, Chapter V]
The problem of Ψ-boundedness of the solutions for systems of ordinary differential equations has been studied in many papers, [1, 2, 3, 5, 9, 10]
Summary
The importance of matrix Lyapunov systems, which arise in a number of areas of control engineering problems, dynamical systems, and feedback systems are well known. The problem of Ψ-boundedness of the solutions for systems of ordinary differential equations has been studied in many papers, [1, 2, 3, 5, 9, 10]. [11, 7], extended the concept of Ψ-boundedness of the solutions to Lyapunov matrix differential equations. In [6], the author obtained necessary and sufficient conditions for the non homogenous system x′ = A(t)x + f (t), to have at least one Ψ-bounded solution on R for every Lebesgue Ψ-integrable function f on R. The aim of present paper is to give a necessary and sufficient condition so that the nonhomogeneous matrix Lyapunov system (1.1) has at least one Ψ-bounded solution on R for every Lebesgue Ψ-integrable.
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