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Boundary Value Problems

, 2010:956121 | Cite as

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

  • J Baštinec
  • J Diblík
  • D Ya Khusainov
  • A Ryvolová
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay:
where Open image in new window is an independent variable, Open image in new window is a constant delay, Open image in new window , and Open image in new window are Open image in new window constant matrices, and Open image in new window is a column vector-solution. The sign " Open image in new window " denotes the left-hand derivative. Let Open image in new window be a continuously differentiable vector-function. The solution Open image in new window of problem (1.1), (1.2) on Open image in new window where

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 .

Let Open image in new window be a rectangular matrix. We will use the matrix norm:
where the symbol Open image in new window denotes the maximal eigenvalue of the corresponding square symmetric positive semidefinite matrix Open image in new window . Similarly, Open image in new window denotes the minimal eigenvalue of Open image in new window . We will use the following vector norms:

where Open image in new window 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

where Open image in new window and Open image in new window are suitable Open image in new window 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

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.

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 Open image in new window 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 Open image in new window and Open image in new window , is studied in [1] where a functional
is used with suitable constants Open image in new window and Open image in new window . 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

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).

In this paper, we will use Lyapunov-Krasovskii quadratic type functionals of the dependent coordinates and their derivatives

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.

The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric Open image in new window if there exist constants Open image in new window , Open image in new window and Open image in new window such that, for an arbitrary solution Open image in new window of (1.1), the inequality

holds for Open image in new window .

Definition 2.2.

The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric Open image in new window if it is stable in the metric Open image in new window and, moreover, there exist constants Open image in new window , Open image in new window , and Open image in new window such that, for an arbitrary solution Open image in new window of (1.1), the inequality

holds for Open image in new window .

We will give estimates of solutions of the linear system (1.1) on the interval Open image in new window using the functional (1.9). Then it is easy to see that an inequality
holds on Open image in new window . We will use an auxiliary Open image in new window -dimensional matrix:
depending on the parameter Open image in new window and the matrices Open image in new window , Open image in new window , Open image in new window . Next we will use the numbers

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.

Let Open image in new window be a positive integer and Open image in new window . Then a solution Open image in new window of the initial problem (1.1), (1.2) is a solution of the delayed system

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 the statement is obvious. If Open image in new window , replacing Open image in new window by Open image in new window , system (1.1) will turn into
Substituting (2.7) into (1.1), we obtain the following system of equations:
We do one more iteration substituting (2.9) into (2.8), obtaining
for Open image in new window . Repeating this procedure Open image in new window -times, we get the equation

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.

Let there exist a parameter Open image in new window and positive definite matrices Open image in new window , Open image in new window , Open image in new window such that matrix Open image in new window is also positive definite. Then the zero solution of system (1.1) is exponentially stable in the metric Open image in new window . Moreover, for the solution Open image in new window of (1.1), (1.2) the inequality

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

Proof.

Let Open image in new window . We will calculate the full derivative of the functional (1.9) along the solutions of system (1.1). We obtain
For Open image in new window , we substitute its value from (1.1) to obtain
Now it is easy to verify that the last expression can be rewritten as
Since the matrix Open image in new window was assumed to be positive definite, for the full derivative of Lyapunov-Krasovskii functional (1.9), we obtain the following inequality:
We will study the two possible cases (depending on the positive value of Open image in new window ): either
is valid or
holds.
  1. (1)
    Let (2.18) be valid. From (2.3) follows that
     
We use this expression in (2.17). Since Open image in new window , we obtain (omitting terms Open image in new window and Open image in new window )
Due to (2.18) we have
Integrating this inequality over the interval Open image in new window , we get
  1. (2)
    Let (2.19) be valid. From (2.3) we get
     
We substitute this expression into inequality (2.17). Since Open image in new window , we obtain (omitting terms Open image in new window and Open image in new window )
Since (2.19) holds, we have
Integrating this inequality over the interval Open image in new window , we get
Combining inequalities (2.24), (2.29), we conclude that, in both cases (2.18), (2.19), we have
and, obviously (see (1.9)),
We use inequality (2.30) to obtain an estimate of the convergence of solutions of system (1.1). From (2.3) follows that
The last inequality implies

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.

Let the matrix Open image in new window be nonsingular and Open image in new window . Let the assumptions of Theorem 2.5 with Open image in new window and Open image in new window be true. Then the zero solution of system (1.1) is exponentially stable in the metric Open image in new window . Moreover, for a solution Open image in new window of (1.1), (1.2), the inequality

holds on Open image in new window .

Proof.

Let Open image in new window . Then the exponential stability of the zero solution in the metric Open image in new window is proved in Theorem 2.5. Now we will show that the zero solution is exponentially stable in the metric Open image in new window as well. As follows from Lemma 2.3, for derivative Open image in new window , the inequality
inequality (2.38) yields
Because Open image in new window , we can estimate
Now we get from (2.40)
the last inequality implies

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

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 Open image in new window gives Open image in new window . For (1.10) the estimate
holds. We define an auxiliary Open image in new window matrix

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.

Theorem 3.1.
  1. (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
     
where Open image in new window .
  1. (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
     

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

Proof.

Let Open image in new window . We compute the full derivative of the functional (1.10) along the solutions of (1.1). For Open image in new window , we substitute its value from (1.1). Finally we get
Since the matrix Open image in new window is positive definite, we have
Now we will study the two possible cases: either
is valid or
holds.
  1. (1)
    Let (3.7) be valid. Since Open image in new window , from inequality (3.1) follows that
     
We use this inequality in (3.6). We obtain
From inequality (3.7) we get
Integrating this inequality over the interval Open image in new window , we get
  1. (2)
    Let (3.8) be valid. From inequality (3.1) we get
     
We use this inequality in (3.6) again. Since Open image in new window , we get
Because the inequality (3.8) holds, we have
Integrating this inequality over the interval Open image in new window , we get
Combining inequalities (3.12), (3.16), we conclude that, in both cases (3.7), (3.8), we have
From this, it follows

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

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.

We will investigate system (1.1) where Open image in new window , Open image in new window ,
that is, the system
with initial conditions (1.2). Set Open image in new window and
and Open image in new window . Because all the eigenvalues are positive, matrix Open image in new window 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 Open image in new window . Further we have
Since Open image in new window , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.2), (35) is asymptotically stable in the metric Open image in new window . Finally, from (2.12) and (2.35) follows that the inequalities

hold on Open image in new window .

Example 4.2.

We will investigate system (1.1) where Open image in new window , Open image in new window ,
that is, the system
with initial conditions (1.2). Set Open image in new window and
and Open image in new window . Because all eigenvalues are positive, matrix Open image in new window 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 Open image in new window . Further we have
Since Open image in new window , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.8) is asymptotically stable in the metric Open image in new window . Finally, from (2.12) and (2.35) follows that the inequalities

hold on Open image in new window .

Remark 4.3.

In [12] an example can be found similar to Example 4.2 with the same matrices Open image in new window , Open image in new window , arbitrary constant positive Open image in new window , and with a matrix

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 .

Comparing these particular results with Example 4.2, we see that, in addition to stability, our results imply the exponential stability in the metric Open image in new window as well as in the metric Open image in new window . Moreover, we are able to prove the exponential stability (in Open image in new window as well as in Open image in new window ) in Example 4.2 with the matrix Open image in new window for Open image in new window and for an arbitrary constant delay Open image in new window . The latter statement can be explained easily—for an arbitrary positive Open image in new window , we set Open image in new window . Calculating the characteristic equation for the matrix Open image in new window where Open image in new window is changed by Open image in new window we get
It is easy to verify that Open image in new window for Open image in new window and Open image in new window , and for the equation

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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Copyright information

© 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.

Authors and Affiliations

  • J Baštinec
    • 1
  • J Diblík
    • 1
    • 2
  • D Ya Khusainov
    • 3
  • A Ryvolová
    • 1
  1. 1.Department of Mathematics, Faculty of Electrical Engineering and CommunicationTechnická 8, Brno University of TechnologyBrnoCzech Republic
  2. 2.Department of Mathematics and Descriptive Geometry, Faculty of Civil EngineeringVeveří 331/95, Brno University of TechnologyBrnoCzech Republic
  3. 3.Department of Complex System Modeling, Faculty of CyberneticsTaras, Shevchenko National University of KyivKyivUkraine

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