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

  • J Baštinec1,

    Affiliated with

    • J Diblík1, 2Email author,

      Affiliated with

      • D Ya Khusainov3 and

        Affiliated with

        • A Ryvolová1

          Affiliated with

          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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ1_HTML.gif
          (1.1)
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq1_HTML.gif is an independent variable, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq2_HTML.gif is a constant delay, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq3_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq4_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq5_HTML.gif constant matrices, and http://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 " http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq7_HTML.gif " denotes the left-hand derivative. Let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq10_HTML.gif where
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq11_HTML.gif continuously differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq12_HTML.gif except for points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq14_HTML.gif , and satisfying (1.1) everywhere on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq15_HTML.gif except for points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq16_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq19_HTML.gif at an arbitrary moment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq20_HTML.gif .

          Let http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ3_HTML.gif
          (1.3)
          where the symbol http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq23_HTML.gif . Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq24_HTML.gif denotes the minimal eigenvalue of http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ4_HTML.gif
          (1.4)

          where http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ5_HTML.gif
          (1.5)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq28_HTML.gif are suitable http://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
          http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq33_HTML.gif and http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq35_HTML.gif and http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ8_HTML.gif
          (1.8)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq38_HTML.gif are positive matrices and http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ9_HTML.gif
          (1.9)
          and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq40_HTML.gif , that is,
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ10_HTML.gif
          (1.10)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq41_HTML.gif is a solution of (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq43_HTML.gif are real parameters, the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq44_HTML.gif matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq46_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq47_HTML.gif are positive definite, and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq49_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq52_HTML.gif if there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq55_HTML.gif such that, for an arbitrary solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq56_HTML.gif of (1.1), the inequality
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ11_HTML.gif
          (2.1)

          holds for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq59_HTML.gif and, moreover, there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq61_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq62_HTML.gif such that, for an arbitrary solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq63_HTML.gif of (1.1), the inequality
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ12_HTML.gif
          (2.2)

          holds for http://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 http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ13_HTML.gif
          (2.3)
          holds on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq66_HTML.gif . We will use an auxiliary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq67_HTML.gif -dimensional matrix:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ14_HTML.gif
          (2.4)
          depending on the parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq68_HTML.gif and the matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq71_HTML.gif . Next we will use the numbers
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq73_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq74_HTML.gif is an integer, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq75_HTML.gif is a positive integer, and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq77_HTML.gif be a positive integer and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq78_HTML.gif . Then a solution http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ16_HTML.gif
          (2.6)

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

          Proof.

          For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq83_HTML.gif the statement is obvious. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq84_HTML.gif , replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq85_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq86_HTML.gif , system (1.1) will turn into
          http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ18_HTML.gif
          (2.8)
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq87_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq88_HTML.gif , replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq89_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq90_HTML.gif in (2.7), we get
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ20_HTML.gif
          (2.10)
          for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq91_HTML.gif . Repeating this procedure http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq92_HTML.gif -times, we get the equation
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ21_HTML.gif
          (2.11)

          for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq94_HTML.gif for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq97_HTML.gif and positive definite matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq100_HTML.gif such that matrix http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq102_HTML.gif . Moreover, for the solution http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ22_HTML.gif
          (2.12)

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

          Proof.

          Let http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ23_HTML.gif
          (2.13)
          For http://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
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ25_HTML.gif
          (2.15)
          or
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ26_HTML.gif
          (2.16)
          Since the matrix http://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:
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq109_HTML.gif ): either
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ28_HTML.gif
          (2.18)
          is valid or
          http://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
            http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq110_HTML.gif , we obtain (omitting terms http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq111_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq112_HTML.gif )
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ31_HTML.gif
          (2.21)
          or
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq113_HTML.gif , we get
          http://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
            http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq114_HTML.gif , we obtain (omitting terms http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq116_HTML.gif )
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ36_HTML.gif
          (2.26)
          or
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq117_HTML.gif , we get
          http://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
          http://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)),
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ42_HTML.gif
          (2.32)
          or (because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq118_HTML.gif for nonnegative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq120_HTML.gif )
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ43_HTML.gif
          (2.33)
          The last inequality implies
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq121_HTML.gif .

          Theorem 2.6.

          Let the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq122_HTML.gif be nonsingular and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq124_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq126_HTML.gif . Moreover, for a solution http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ45_HTML.gif
          (2.35)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ46_HTML.gif
          (2.36)

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

          Proof.

          Let http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq132_HTML.gif , the inequality
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ47_HTML.gif
          (2.37)
          holds if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq133_HTML.gif We estimate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq135_HTML.gif using (2.12) and inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq136_HTML.gif . We obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ48_HTML.gif
          (2.38)
          Since
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ49_HTML.gif
          (2.39)
          inequality (2.38) yields
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ50_HTML.gif
          (2.40)
          Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq137_HTML.gif , we can estimate
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ51_HTML.gif
          (2.41)
          Then
          http://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)
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ53_HTML.gif
          (2.43)
          Since
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ54_HTML.gif
          (2.44)
          the last inequality implies
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ55_HTML.gif
          (2.45)

          The positive number http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq141_HTML.gif gives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq142_HTML.gif . For (1.10) the estimate
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq143_HTML.gif matrix
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ57_HTML.gif
          (3.2)

          depending on the parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq144_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq145_HTML.gif and the matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq147_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq148_HTML.gif . The parameter http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq150_HTML.gif . Particularly, a proper choice of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq151_HTML.gif can cause the positivity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq152_HTML.gif . In the following, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq154_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq157_HTML.gif be a fixed real number, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq158_HTML.gif a positive constant, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq159_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq160_HTML.gif , and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq164_HTML.gif the inequality
            http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ58_HTML.gif
            (3.3)
             
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq165_HTML.gif .
          1. (B)
            Let the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq166_HTML.gif be nonsingular and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq168_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq171_HTML.gif the inequality
            http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ59_HTML.gif
            (3.4)
             

          where http://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 http://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 http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ60_HTML.gif
          (3.5)
          Since the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq175_HTML.gif is positive definite, we have
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ62_HTML.gif
          (3.7)
          is valid or
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq176_HTML.gif , from inequality (3.1) follows that
            http://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
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq177_HTML.gif , we get
          http://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
            http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq178_HTML.gif , we get
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq179_HTML.gif , we get
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ72_HTML.gif
          (3.17)
          From this, it follows
          http://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
          http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq183_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ75_HTML.gif
          (4.1)
          that is, the system
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq184_HTML.gif and
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq185_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq186_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq187_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq190_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq191_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq192_HTML.gif . The matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq193_HTML.gif takes the form
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ78_HTML.gif
          (4.4)
          and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq196_HTML.gif . Further we have
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ79_HTML.gif
          (4.5)
          Since http://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 http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ80_HTML.gif
          (4.6)

          hold on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq200_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq201_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ81_HTML.gif
          (4.7)
          that is, the system
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq202_HTML.gif and
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq203_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq204_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq205_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq208_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq209_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq210_HTML.gif . The matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq211_HTML.gif takes the form
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ84_HTML.gif
          (4.10)
          and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq212_HTML.gif . Because all eigenvalues are positive, matrix http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq214_HTML.gif . Further we have
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ85_HTML.gif
          (4.11)
          Since http://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 http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ86_HTML.gif
          (4.12)

          hold on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq219_HTML.gif , arbitrary constant positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq220_HTML.gif , and with a matrix
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ87_HTML.gif
          (4.13)

          where http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq224_HTML.gif as well as in the metric http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq226_HTML.gif as well as in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq228_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq229_HTML.gif and for an arbitrary constant delay http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq231_HTML.gif , we set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq233_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq234_HTML.gif is changed by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq235_HTML.gif we get
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ88_HTML.gif
          (4.14)
          where
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq236_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq237_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq238_HTML.gif , and for the equation
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_Equ90_HTML.gif
          (4.16)

          we have http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq241_HTML.gif (i.e., all roots of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq243_HTML.gif as well as in the metric http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq244_HTML.gif ) for http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq248_HTML.gif and http://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 http://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 http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq253_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F956121/MediaObjects/13661_2010_Article_969_IEq255_HTML.gif ) and to a variable delay ( http://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

          References

          1. Kolmanovskii V, Myshkis A: Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, the Netherlands; 1999:xvi+648.View Article
          2. Krasovskii NN: Some Problems of Theory of Stability of Motion. Fizmatgiz, Moscow, Russia; 1959.
          3. Krasovskiĭ NN: Stability of Motion. Applications of Lyapunov's Second Method to Differential systems and Equations with Delay, Translated by J. L. Brenner. Stanford University Press, Stanford, Calif, USA; 1963:vi+188.
          4. Korenevskiĭ DG: Stability of Dynamical Systems Under Random Perturbations of Parameters. Algebraic Criteria. Naukova Dumka, Kiev, Ukraine; 1989:208.
          5. Khusainov DYa, Yunkova EA: Investigation of the stability of linear systems of neutral type by the method of Lyapunov functions. Differentsial cprime nye Uravneniya 1988,24(4):613-621. Translated in Differential Equations, vol. 24, no. 4, pp. 424–431MathSciNet
          6. Gopalsamy K: Stability and Oscillations in Delay Differential Equations of Population Pynamics, Mathematics and its Applications. Volume 74. Kluwer Academic Publishers, Dordrecht, the Netherlands; 1992:xii+501.View Article
          7. Kolmanovskiĭ V, Myshkis A: Applied Theory of Functional-Differential Equations, Mathematics and its Applications (Soviet Series). Volume 85. Kluwer Academic Publishers Group, Dordrecht, the Netherlands; 1992:xvi+234.View Article
          8. Kolmanovskiĭ V, Nosov V: Stability of Functional Differential Equations, Mathematics in Science and Engineering. Volume 180. Academic Press, Harcourt Brace Jovanovich, London, UK; 1986:xiv+ 217.
          9. Mei-Gin L: Stability analysis of neutral-type nonlinear delayed systems: an LMI approach. Journal of Zhejiang University A 2006, 7, supplement 2: 237-244.
          10. Gu K, Kharitonov VL, Chen J: Stability of Time-Delay Systems, Control Engineering. Birkhuser, Boston, Mass, USA; 2003:xx+353.View Article
          11. Liao X, Wang L, Yu P: Stability of Dynamical Systems, Monograph Series on Nonlinear Science and Complexity. Volume 5. Elsevier, Amsterdam, the Netherlands; 2007:xii+706.
          12. Park Ju-H, Won S: A note on stability of neutral delay-differential systems. Journal of the Franklin Institute 1999,336(3):543-548. 10.1016/S0016-0032(98)00040-4MathSciNetView Article
          13. Liu X-x, Xu B: A further note on stability criterion of linear neutral delay-differential systems. Journal of the Franklin Institute 2006,343(6):630-634. 10.1016/j.jfranklin.2006.02.024MathSciNetView Article

          Copyright

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

          This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.