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

## 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

Exponential Stability Full Derivative Zero Solution Neutral System Constant Delay## 1. Introduction

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

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

where Open image in new window is a parameter.

where Open image in new window and Open image in new window are suitable Open image in new window positive definite matrices.

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

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

holds for Open image in new window .

Definition 2.2.

holds for Open image in new window .

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

Lemma 2.3.

for Open image in new window where Open image in new window and Open image in new window .

Proof.

for Open image in new window 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 Open image in new window for Open image in new window depending only on the derivative of the initial function on the initial interval Open image in new window .

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.

holds on Open image in new window where Open image in new window .

Proof.

- (1)Let (2.18) be valid. From (2.3) follows that(2.20)

- (2)Let (2.19) be valid. From (2.3) we get(2.25)

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

Theorem 2.6.

holds on Open image in new window .

Proof.

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

## 3. Estimates of Solutions in a General Case

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

- (A)Let Open image in new window be a fixed real number, Open image in new window a positive constant, and Open image in new window , Open image in new window , and Open image in new window positive definite matrices such that the matrix Open image in new window is also positive definite. Then a solution Open image in new window of problem (1.1), (1.2) satisfies on Open image in new window the inequality(3.3)

- (B)Let the matrix Open image in new window be nonsingular and Open image in new window . Let all the assumptions of part (A) with Open image in new window and Open image in new window be true. Then the derivative of the solution Open image in new window of problem (1.1), (1.2) satisfies on Open image in new window the inequality(3.4)

where Open image in new window is defined by (2.36).

Proof.

- (1)Let (3.7) be valid. Since Open image in new window , from inequality (3.1) follows that(3.9)

- (2)Let (3.8) be valid. From inequality (3.1) we get(3.13)

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

## 4. Examples

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

Example 4.1.

hold on Open image in new window .

Example 4.2.

hold on Open image in new window .

Remark 4.3.

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

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

## 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 Open image in new window , 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 Open image in new window 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 Open image in new window and estimates of the norm of a derivative of solution). Such an approach will probably permit a generalization to variable matrices ( Open image in new window , Open image in new window , Open image in new window ) and to a variable delay ( Open image in new window ) or to two different variable delays. Nevertheless, it seems that the results obtained will be very cumbersome and hardly applicable in practice.

## Notes

### 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.

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