- Research Article
- Open Access

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

- J Baštinec
^{1}, - J Diblík
^{1, 2}Email author, - D Ya Khusainov
^{3}and - A Ryvolová
^{1}

**2010**:956121

https://doi.org/10.1155/2010/956121

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

**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.

## Keywords

## 1. Introduction

is defined in the classical sense (we refer, e.g., to [1]) as a function continuous on continuously differentiable on except for points , , and satisfying (1.1) everywhere on except for points , .

The paper finds an estimate of the norm of the difference between a solution of problem (1.1), (1.2) and the steady state at an arbitrary moment .

where and are suitable positive definite matrices.

is bounded from above are of an integral type. Because the terms in (1.5) contain differences, they do not imply the boundedness of the norm of itself.

where and are positive matrices and 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).

where is a solution of (1.1), and are real parameters, the matrices , , and are positive definite, and . 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 and even in the case of instability. An estimate of the norm 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.

Definition 2.2.

The following lemma gives a representation of the linear neutral system (1.1) on an interval in terms of a delayed system derived by an iterative process. We will adopt the customary notation where is an integer, is a positive integer, and denotes the function considered independently of whether it is defined for the arguments indicated or not.

Lemma 2.3.

Proof.

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 for depending only on the derivative of the initial function on the initial interval .

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.

Proof.

Thus inequality (2.12) is proved and, consequently, the zero solution of system (1.1) is exponentially stable in the metric .

Theorem 2.6.

Proof.

The positive number can be chosen arbitrarily large. Therefore, the last inequality holds for every . We have obtained inequality (2.35) so that the zero solution of (1.1) is exponentially stable in the metric .

## 3. Estimates of Solutions in a General Case

depending on the parameters , and the matrices , , and . The parameter plays a significant role for the positive definiteness of the matrix . Particularly, a proper choice of can cause the positivity of . In the following, , and , 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 exactly coincides with the statements of these theorems). Therefore, we will restrict its proof to the main points only.

- (A)

Proof.

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.

we deal with an exponential stability in the metric . If, moreover, part (B) holds and (3.19) is valid, then we deal with an exponential stability in the metric .

## 4. Examples

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

Example 4.1.

Example 4.2.

Remark 4.3.

where is a real parameter. The stability is established for . In recent paper [13], the stability of the same system is even established for .

we have . Then, due to the symmetry of the real matrix , we conclude that, by Descartes' rule of signs, all eigenvalues of (i.e., all roots of ) are positive. This means that the exponential stability (in the metric as well as in the metric ) for is proved. Finally, we note that the variation of within the interval indicated or the choice does not change the exponential stability having only influence on the form of the final inequalities for and .

## 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 , 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 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 and estimates of the norm of a derivative of solution). Such an approach will probably permit a generalization to variable matrices ( , , ) and to a variable delay ( ) 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

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