Open Access

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

Boundary Value Problems20102010:956121

DOI: 10.1155/2010/956121

Received: 6 July 2010

Accepted: 12 October 2010

Published: 18 October 2010

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.

1. Introduction

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq1_HTML.gif is an independent variable, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq2_HTML.gif is a constant delay, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq4_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq5_HTML.gif constant matrices, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq6_HTML.gif is a column vector-solution. The sign " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq7_HTML.gif " denotes the left-hand derivative. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq8_HTML.gif be a continuously differentiable vector-function. The solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq9_HTML.gif of problem (1.1), (1.2) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq10_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ2_HTML.gif
(1.2)

is defined in the classical sense (we refer, e.g., to [1]) as a function continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq11_HTML.gif continuously differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq12_HTML.gif except for points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq14_HTML.gif , and satisfying (1.1) everywhere on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq15_HTML.gif except for points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq17_HTML.gif .

The paper finds an estimate of the norm of the difference between a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq18_HTML.gif of problem (1.1), (1.2) and the steady state https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq19_HTML.gif at an arbitrary moment https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq20_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq21_HTML.gif be a rectangular matrix. We will use the matrix norm:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ3_HTML.gif
(1.3)
where the symbol https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq22_HTML.gif denotes the maximal eigenvalue of the corresponding square symmetric positive semidefinite matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq23_HTML.gif . Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq24_HTML.gif denotes the minimal eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq25_HTML.gif . We will use the following vector norms:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq26_HTML.gif is a parameter.

The most frequently used method for investigating the stability of functional-differential systems is the method of Lyapunov-Krasovskii functionals [2, 3]. Usually, it uses positive definite functionals of a quadratic form generated from terms of (1.1) and the integral (over the interval of delay [4]) of a quadratic form. A possible form of such a functional is then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq28_HTML.gif are suitable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq29_HTML.gif positive definite matrices.

Regarding the functionals of the form (1.5), we should underline the following. Using a functional (1.5), we can only obtain propositions concerning the stability. Statements such as that the expression
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ6_HTML.gif
(1.6)

is bounded from above are of an integral type. Because the terms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq30_HTML.gif in (1.5) contain differences, they do not imply the boundedness of the norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq31_HTML.gif itself.

Literature on the stability and estimation of solutions of neutral differential equations is enormous. Tracing previous investigations on this topic, we emphasize that a Lyapunov function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq32_HTML.gif has been used to investigate the stability of systems (1.1) in [5] (see [6] as well). The stability of linear neutral systems of type (1.1), but with different delays https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq34_HTML.gif , is studied in [1] where a functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ7_HTML.gif
(1.7)
is used with suitable constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq36_HTML.gif . In [7, 8], functionals depending on derivatives are also 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ8_HTML.gif
(1.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq38_HTML.gif are positive matrices and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq39_HTML.gif is a positive scalar.

Delay independent criteria of stability for some classes of delay neutral systems are developed in [10]. The stability of systems (1.1) with time dependent delays is investigated in [11]. For recent results on the stability of neutral equations, see [9, 12] and the references therein. The works in [12, 13] deal with delay independent criteria of the asymptotical stability of systems (1.1).

In this paper, we will use Lyapunov-Krasovskii quadratic type functionals of the dependent coordinates and their derivatives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ9_HTML.gif
(1.9)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq40_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ10_HTML.gif
(1.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq41_HTML.gif is a solution of (1.1), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq43_HTML.gif are real parameters, the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq44_HTML.gif matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq46_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq47_HTML.gif are positive definite, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq48_HTML.gif . The form of functionals (1.9) and (1.10) is suggested by the functionals (1.7)-(1.8). Although 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, also gives delay-dependent estimates of solutions in terms of the norms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq50_HTML.gif even in the case of instability. An estimate of the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq51_HTML.gif can be achieved by reducing the initial neutral system (1.1) to a neutral system having the same solution on the intervals indicated in which the "neutrality" is concentrated only on the initial interval. If, in the literature, estimates of solutions are given, then, as a rule, estimates of derivatives are not investigated.

To the best of our knowledge, the general functionals (1.9) and (1.10) have not yet been applied as suggested to the study of stability and estimates of solutions of (1.1).

2. Exponential Stability and Estimates of the Convergence of Solutions to Stable Systems

First we give two definitions of stability to be used later on.

Definition 2.1.

The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq52_HTML.gif if there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq55_HTML.gif such that, for an arbitrary solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq56_HTML.gif of (1.1), the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ11_HTML.gif
(2.1)

holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq57_HTML.gif .

Definition 2.2.

The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq58_HTML.gif if it is stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq59_HTML.gif and, moreover, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq61_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq62_HTML.gif such that, for an arbitrary solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq63_HTML.gif of (1.1), the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ12_HTML.gif
(2.2)

holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq64_HTML.gif .

We will give estimates of solutions of the linear system (1.1) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq65_HTML.gif using the functional (1.9). Then it is easy to see that an inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ13_HTML.gif
(2.3)
holds on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq66_HTML.gif . We will use an auxiliary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq67_HTML.gif -dimensional matrix:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ14_HTML.gif
(2.4)
depending on the parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq68_HTML.gif and the matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq71_HTML.gif . Next we will use the numbers
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ15_HTML.gif
(2.5)

The following lemma gives a representation of the linear neutral system (1.1) on an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq72_HTML.gif in terms of a delayed system derived by an iterative process. We will adopt the customary notation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq73_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq74_HTML.gif is an integer, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq75_HTML.gif is a positive integer, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq76_HTML.gif denotes the function considered independently of whether it is defined for the arguments indicated or not.

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq77_HTML.gif be a positive integer and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq78_HTML.gif . Then a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq79_HTML.gif of the initial problem (1.1), (1.2) is a solution of the delayed system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ16_HTML.gif
(2.6)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq80_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq82_HTML.gif .

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq83_HTML.gif the statement is obvious. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq84_HTML.gif , replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq85_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq86_HTML.gif , system (1.1) will turn into
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ17_HTML.gif
(2.7)
Substituting (2.7) into (1.1), we obtain the following system of equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ18_HTML.gif
(2.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq87_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq88_HTML.gif , replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq89_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq90_HTML.gif in (2.7), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ19_HTML.gif
(2.9)
We do one more iteration substituting (2.9) into (2.8), obtaining
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ20_HTML.gif
(2.10)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq91_HTML.gif . Repeating this procedure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq92_HTML.gif -times, we get the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ21_HTML.gif
(2.11)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq93_HTML.gif coinciding with (2.6).

Remark 2.4.

The advantage of representing a solution of the initial problem (1.1), (1.2) as a solution of (2.6) is that, although (2.6) remains to be a neutral system, its right-hand side does not explicitly depend on the derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq94_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq95_HTML.gif depending only on the derivative of the initial function on the initial interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq96_HTML.gif .

Now we give a statement on the stability of the zero solution of system (1.1) and estimates of the convergence of the solution, which we will prove using Lyapunov-Krasovskii functional (1.9).

Theorem 2.5.

Let there exist a parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq97_HTML.gif and positive definite matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq100_HTML.gif such that matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq101_HTML.gif is also positive definite. Then the zero solution of system (1.1) is exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq102_HTML.gif . Moreover, for the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq103_HTML.gif of (1.1), (1.2) the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ22_HTML.gif
(2.12)

holds on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq104_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq105_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq106_HTML.gif . We will calculate the full derivative of the functional (1.9) along the solutions of system (1.1). We obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ23_HTML.gif
(2.13)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq107_HTML.gif , we substitute its value from (1.1) to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ24_HTML.gif
(2.14)
Now it is easy to verify that the last expression can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ25_HTML.gif
(2.15)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ26_HTML.gif
(2.16)
Since the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq108_HTML.gif was assumed to be positive definite, for the full derivative of Lyapunov-Krasovskii functional (1.9), we obtain the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ27_HTML.gif
(2.17)
We will study the two possible cases (depending on the positive value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq109_HTML.gif ): either
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ28_HTML.gif
(2.18)
is valid or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ29_HTML.gif
(2.19)
holds.
  1. (1)
    Let (2.18) be valid. From (2.3) follows that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ30_HTML.gif
    (2.20)
     
We use this expression in (2.17). Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq110_HTML.gif , we obtain (omitting terms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq112_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ31_HTML.gif
(2.21)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ32_HTML.gif
(2.22)
Due to (2.18) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ33_HTML.gif
(2.23)
Integrating this inequality over the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq113_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ34_HTML.gif
(2.24)
  1. (2)
    Let (2.19) be valid. From (2.3) we get
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ35_HTML.gif
    (2.25)
     
We substitute this expression into inequality (2.17). Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq114_HTML.gif , we obtain (omitting terms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq116_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ36_HTML.gif
(2.26)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ37_HTML.gif
(2.27)
Since (2.19) holds, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ38_HTML.gif
(2.28)
Integrating this inequality over the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq117_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ39_HTML.gif
(2.29)
Combining inequalities (2.24), (2.29), we conclude that, in both cases (2.18), (2.19), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ40_HTML.gif
(2.30)
and, obviously (see (1.9)),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ41_HTML.gif
(2.31)
We use inequality (2.30) to obtain an estimate of the convergence of solutions of system (1.1). From (2.3) follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ42_HTML.gif
(2.32)
or (because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq118_HTML.gif for nonnegative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq120_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ43_HTML.gif
(2.33)
The last inequality implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ44_HTML.gif
(2.34)

Thus inequality (2.12) is proved and, consequently, the zero solution of system (1.1) is exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq121_HTML.gif .

Theorem 2.6.

Let the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq122_HTML.gif be nonsingular and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq123_HTML.gif . Let the assumptions of Theorem 2.5 with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq125_HTML.gif be true. Then the zero solution of system (1.1) is exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq126_HTML.gif . Moreover, for a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq127_HTML.gif of (1.1), (1.2), the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ45_HTML.gif
(2.35)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ46_HTML.gif
(2.36)

holds on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq128_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq129_HTML.gif . Then the exponential stability of the zero solution in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq130_HTML.gif is proved in Theorem 2.5. Now we will show that the zero solution is exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq131_HTML.gif as well. As follows from Lemma 2.3, for derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq132_HTML.gif , the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ47_HTML.gif
(2.37)
holds if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq133_HTML.gif We estimate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq135_HTML.gif using (2.12) and inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq136_HTML.gif . We obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ48_HTML.gif
(2.38)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ49_HTML.gif
(2.39)
inequality (2.38) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ50_HTML.gif
(2.40)
Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq137_HTML.gif , we can estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ51_HTML.gif
(2.41)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ52_HTML.gif
(2.42)
Now we get from (2.40)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ53_HTML.gif
(2.43)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ54_HTML.gif
(2.44)
the last inequality implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ55_HTML.gif
(2.45)

The positive number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq138_HTML.gif can be chosen arbitrarily large. Therefore, the last inequality holds for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq139_HTML.gif . We have obtained inequality (2.35) so that the zero solution of (1.1) is exponentially stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq140_HTML.gif .

3. Estimates of Solutions in a General Case

Now we will estimate the norms of solutions of (1.1) and the norms of their derivatives in the case of the assumptions of Theorem 2.5 or Theorem 2.6 being not necessarily satisfied. It means that the estimates derived will cover the case of instability as well. For obtaining such type of results we will use a functional of Lyapunov-Krasovskii in the form (1.10). This functional includes an exponential factor, which makes it possible, in the case of instability, to get an estimate of the "divergence" of solutions. Functional (1.10) is a generalization of (1.9) because the choice https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq141_HTML.gif gives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq142_HTML.gif . For (1.10) the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ56_HTML.gif
(3.1)
holds. We define an auxiliary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq143_HTML.gif matrix
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ57_HTML.gif
(3.2)

depending on the parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq144_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq145_HTML.gif and the matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq146_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq147_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq148_HTML.gif . The parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq149_HTML.gif plays a significant role for the positive definiteness of the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq150_HTML.gif . Particularly, a proper choice of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq151_HTML.gif can cause the positivity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq152_HTML.gif . In the following, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq155_HTML.gif , have the same meaning as in Part 2. The proof of the following theorem is similar to the proofs of Theorems 2.5 and 2.6 (and its statement in the case of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq156_HTML.gif exactly coincides with the statements of these theorems). Therefore, we will restrict its proof to the main points only.

Theorem 3.1.
  1. (A)
    Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq157_HTML.gif be a fixed real number, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq158_HTML.gif a positive constant, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq159_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq160_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq161_HTML.gif positive definite matrices such that the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq162_HTML.gif is also positive definite. Then a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq163_HTML.gif of problem (1.1), (1.2) satisfies on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq164_HTML.gif the inequality
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ58_HTML.gif
    (3.3)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq165_HTML.gif .
  1. (B)
    Let the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq166_HTML.gif be nonsingular and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq167_HTML.gif . Let all the assumptions of part (A) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq169_HTML.gif be true. Then the derivative of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq170_HTML.gif of problem (1.1), (1.2) satisfies on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq171_HTML.gif the inequality
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ59_HTML.gif
    (3.4)
     

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq172_HTML.gif is defined by (2.36).

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq173_HTML.gif . We compute the full derivative of the functional (1.10) along the solutions of (1.1). For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq174_HTML.gif , we substitute its value from (1.1). Finally we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ60_HTML.gif
(3.5)
Since the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq175_HTML.gif is positive definite, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ61_HTML.gif
(3.6)
Now we will study the two possible cases: either
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ62_HTML.gif
(3.7)
is valid or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ63_HTML.gif
(3.8)
holds.
  1. (1)
    Let (3.7) be valid. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq176_HTML.gif , from inequality (3.1) follows that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ64_HTML.gif
    (3.9)
     
We use this inequality in (3.6). We obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ65_HTML.gif
(3.10)
From inequality (3.7) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ66_HTML.gif
(3.11)
Integrating this inequality over the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq177_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ67_HTML.gif
(3.12)
  1. (2)
    Let (3.8) be valid. From inequality (3.1) we get
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ68_HTML.gif
    (3.13)
     
We use this inequality in (3.6) again. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq178_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ69_HTML.gif
(3.14)
Because the inequality (3.8) holds, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ70_HTML.gif
(3.15)
Integrating this inequality over the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq179_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ71_HTML.gif
(3.16)
Combining inequalities (3.12), (3.16), we conclude that, in both cases (3.7), (3.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ72_HTML.gif
(3.17)
From this, it follows
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ73_HTML.gif
(3.18)

From the last inequality we derive inequality (3.3) in a way similar to that of the proof of Theorem 2.5. The inequality to estimate the derivative (3.4) can be obtained in much the same way as in the proof of Theorem 2.6.

Remark 3.2.

As can easily be seen from Theorem 3.1, part (A), if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ74_HTML.gif
(3.19)

we deal with an exponential stability in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq180_HTML.gif . If, moreover, part (B) holds and (3.19) is valid, then we deal with an exponential stability in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq181_HTML.gif .

4. Examples

In this part we consider two examples. Auxiliary numerical computations were performed by using MATLAB & SIMULINK R2009a.

Example 4.1.

We will investigate system (1.1) where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq183_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ75_HTML.gif
(4.1)
that is, the system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ76_HTML.gif
(4.2)
with initial conditions (1.2). Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq184_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ77_HTML.gif
(4.3)
For the eigenvalues of matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq186_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq187_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq190_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq191_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq192_HTML.gif . The matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq193_HTML.gif takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ78_HTML.gif
(4.4)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq194_HTML.gif . Because all the eigenvalues are positive, matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq195_HTML.gif is positive definite. Since all conditions of Theorem 2.5 are satisfied, the zero solution of system (4.2) is asymptotically stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq196_HTML.gif . Further we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ79_HTML.gif
(4.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq197_HTML.gif , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.2), (35) is asymptotically stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq198_HTML.gif . Finally, from (2.12) and (2.35) follows that the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ80_HTML.gif
(4.6)

hold on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq199_HTML.gif .

Example 4.2.

We will investigate system (1.1) where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq201_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ81_HTML.gif
(4.7)
that is, the system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ82_HTML.gif
(4.8)
with initial conditions (1.2). Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq202_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ83_HTML.gif
(4.9)
For the eigenvalues of matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq203_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq204_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq205_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq206_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq207_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq208_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq209_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq210_HTML.gif . The matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq211_HTML.gif takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ84_HTML.gif
(4.10)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq212_HTML.gif . Because all eigenvalues are positive, matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq213_HTML.gif is positive definite. Since all conditions of Theorem 2.5 are satisfied, the zero solution of system (4.8) is asymptotically stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq214_HTML.gif . Further we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ85_HTML.gif
(4.11)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq215_HTML.gif , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.8) is asymptotically stable in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq216_HTML.gif . Finally, from (2.12) and (2.35) follows that the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ86_HTML.gif
(4.12)

hold on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq217_HTML.gif .

Remark 4.3.

In [12] an example can be found similar to Example 4.2 with the same matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq219_HTML.gif , arbitrary constant positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq220_HTML.gif , and with a matrix
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ87_HTML.gif
(4.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq221_HTML.gif is a real parameter. The stability is established for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq222_HTML.gif . In recent paper [13], the stability of the same system is even established for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq223_HTML.gif .

Comparing these particular results with Example 4.2, we see that, in addition to stability, our results imply the exponential stability in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq224_HTML.gif as well as in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq225_HTML.gif . Moreover, we are able to prove the exponential stability (in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq226_HTML.gif as well as in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq227_HTML.gif ) in Example 4.2 with the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq228_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq229_HTML.gif and for an arbitrary constant delay https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq230_HTML.gif . The latter statement can be explained easily—for an arbitrary positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq231_HTML.gif , we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq232_HTML.gif . Calculating the characteristic equation for the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq233_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq234_HTML.gif is changed by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq235_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ88_HTML.gif
(4.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ89_HTML.gif
(4.15)
It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq236_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq238_HTML.gif , and for the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ90_HTML.gif
(4.16)

we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq239_HTML.gif . Then, due to the symmetry of the real matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq240_HTML.gif , we conclude that, by Descartes' rule of signs, all eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq241_HTML.gif (i.e., all roots of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq242_HTML.gif ) are positive. This means that the exponential stability (in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq243_HTML.gif as well as in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq244_HTML.gif ) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq245_HTML.gif is proved. Finally, we note that the variation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq246_HTML.gif within the interval indicated or the choice https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq247_HTML.gif does not change the exponential stability having only influence on the form of the final inequalities for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq248_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq249_HTML.gif .

5. Conclusions

In this paper we derived statements on the exponential stability of system (1.1) as well as on estimates of the norms of its solutions and their derivatives in the case of exponential stability and in the case of exponential stability being not guaranteed. To obtain these results, 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). The flexibility and power of this method was demonstrated using examples and comparisons with other results in this field. Considering further possibilities along these lines, we conclude that, to generalize the results presented to systems with bounded variable delay https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq250_HTML.gif , a generalization is needed of Lemma 2.3 to the above reduced neutral system. This can cause substantial difficulties in obtaining results which are easily presentable. An alternative would be to generalize only the part of the results related to the exponential stability in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq251_HTML.gif and the related estimates of the norms of solutions in the case of exponential stability and in the case of the exponential stability being not guaranteed (omitting the case of exponential stability in the metric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq252_HTML.gif and estimates of the norm of a derivative of solution). Such an approach will probably permit a generalization to variable matrices ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq253_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq255_HTML.gif ) and to a variable delay ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq256_HTML.gif ) or to two different variable delays. Nevertheless, it seems that the results obtained will be very cumbersome and hardly applicable in practice.

Declarations

Acknowledgments

J. Baštinec was supported by Grant 201/10/1032 of Czech Grant Agency, by the Council of Czech Government MSM 0021630529, and by Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. J. Diblík was supported by Grant 201/08/9469 of Czech Grant Agency, by the Council of Czech Government MSM 0021630503, MSM 0021630519, and by Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. D. Ya. Khusainov was supported by project M/34-2008 MOH Ukraine since March 27, 2008. A. Ryvolová was supported by the Council of Czech Government MSM 0021630529, and by Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Electrical Engineering and Communication, Technická 8, Brno University of Technology
(2)
Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Veveří 331/95, Brno University of Technology
(3)
Department of Complex System Modeling, Faculty of Cybernetics, Taras, Shevchenko National University of Kyiv

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© The Author(s) J. Baštinec et al. 2010

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