On Stability of Linear Delay Differential Equations under Perron's Condition

J Diblík,A Zafer

doi:10.1155/2011/134072

Abstract

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established.

Highlights

We begin with a classical result for the linear system x A t x, L1 where A is an n × n matrix function defined and continuous on 0, ∞
Theorem 1.2 is extended by Halanay 5 to linear delay systems of the form xtAtxtBtxt−τ, L2 where A, B are n × n matrix functions defined and continuous on 0, ∞ and τ is a positive real number
For a given continuous vector function φ defined on g t0, t0, let x t x t, t0, φ denote the solution of 1.4 satisfying x t φ t, t ≤ t0

Summary

Introduction

We begin with a classical result for the linear system x A t x, L1 where A is an n × n matrix function defined and continuous on 0, ∞. The proof is accomplished by making use of the basic properties of a fundamental matrix, the Banach-Steinhaus theorem, and the adjoint system x −AT t x, 1.1 where AT denotes the transpose of A It is shown by an example in 3 that Theorem 1.2 may not be valid if the function f appearing in N1 is replaced by a constant vector. Theorem 1.2 is extended by Halanay 5 to linear delay systems of the form xtAtxtBtxt−τ , L2 where A, B are n × n matrix functions defined and continuous on 0, ∞ and τ is a positive real number. System 1.4 is said to satisfy Perron’s condition P if, for any given vector function f ∈ CB 0, ∞ , the solution x t of x t Atxt Btx gt f t.

Stability Theorems

Variation of Parameters Formula

Auxiliary Results

Proofs of Theorems