Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients

J Baštinec,A Ryvolová,J Diblík,Dya Khusainov

doi:10.1155/2010/956121

Abstract

This paper investigates the exponential-type stability of linear neutral delay differential systems with constant coefficients using Lyapunov-Krasovskii type functionals, more general than those reported in the literature. Delay-dependent conditions sufficient for the stability are formulated in terms of positivity of auxiliary matrices. The approach developed is used to characterize the decay of solutions (by inequalities for the norm of an arbitrary solution and its derivative) in the case of stability, as well as in a general case. Illustrative examples are shown and comparisons with known results are given.

Highlights

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay: xt Dxt − τ Ax t Bx t − τ, 1.1 where t ≥ 0 is an independent variable, τ > 0 is a constant delay, A, B, and D are n×n constant matrices, and x : −τ, ∞ → Rn is a column vector-solution
This paper investigates the exponential-type stability of linear neutral delay differential systems with constant coefficients using Lyapunov-Krasovskii type functionals, more general than those reported in the literature
Special Lyapunov functionals in the form 1.9 and 1.10 were utilized as well as a method of constructing a reduced neutral system with the same solution on the intervals indicated as the initial neutral system 1.1

Summary

Introduction

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay: xt Dxt − τ Ax t Bx t − τ , 1.1 where t ≥ 0 is an independent variable, τ > 0 is a constant delay, A, B, and D are n×n constant matrices, and x : −τ, ∞ → Rn is a column vector-solution. In 7, 8 , functionals depending on derivatives are suggested for investigating the asymptotic stability of neutral nonlinear systems. The investigation of nonlinear neutral delayed systems with two time dependent bounded delays in 9 to determine the global asymptotic and exponential stability uses, for example, functionals xT t P x t xT t s Qx s ds x T t s xt s ds,. Many approaches in the literature are used to judge the stability, our approach, among others, in addition to determining whether the system 1.1 is exponentially stable, gives delay-dependent estimates of solutions in terms of the norms x t and xt even in the case of instability. Since the matrix S was assumed to be positive definite, for the full derivative of LyapunovKrasovskii functional 1.9 , we obtain the following inequality:. M φH τ φ1 G1, H x 0 τ τ φ2 G2, H x 0 τ e−γt/2

Estimates of Solutions in a General Case

Examples

Conclusions