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

This paper will provide estimates of solutions of linear systems of neutral differential equations with constant coefficients and a constant delay:

(1.1)

where is an independent variable, is a constant delay, , and are constant matrices, and is a column vector-solution. The sign " " denotes the left-hand derivative. Let be a continuously differentiable vector-function. The solution of problem (1.1), (1.2) on where

(1.2)

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 .

Let be a rectangular matrix. We will use the matrix norm:

(1.3)

where the symbol denotes the maximal eigenvalue of the corresponding square symmetric positive semidefinite matrix . Similarly, denotes the minimal eigenvalue of . We will use the following vector norms:

(1.4)

where 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

(1.5)

where and are suitable 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

(1.6)

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.

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 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 and , is studied in [1] where a functional

(1.7)

is used with suitable constants and . 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

(1.8)

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).

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

(1.9)

and , that is,

(1.10)

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).

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 if there exist constants , and such that, for an arbitrary solution of (1.1), the inequality

(2.1)

holds for .

Definition 2.2.

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

(2.2)

holds for .

We will give estimates of solutions of the linear system (1.1) on the interval using the functional (1.9). Then it is easy to see that an inequality

(2.3)

holds on . We will use an auxiliary -dimensional matrix:

(2.4)

depending on the parameter and the matrices , , . Next we will use the numbers

(2.5)

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.

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

(2.6)

for where and .

Proof.

For the statement is obvious. If , replacing by , system (1.1) will turn into

(2.7)

Substituting (2.7) into (1.1), we obtain the following system of equations:

(2.8)

where . If , replacing by in (2.7), we get

(2.9)

We do one more iteration substituting (2.9) into (2.8), obtaining

(2.10)

for . Repeating this procedure -times, we get the equation

(2.11)

for 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 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.

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

(2.12)

holds on where .

Proof.

Let . We will calculate the full derivative of the functional (1.9) along the solutions of system (1.1). We obtain

(2.13)

For , we substitute its value from (1.1) to obtain

(2.14)

Now it is easy to verify that the last expression can be rewritten as

(2.15)

or

(2.16)

Since the matrix was assumed to be positive definite, for the full derivative of Lyapunov-Krasovskii functional (1.9), we obtain the following inequality:

(2.17)

We will study the two possible cases (depending on the positive value of ): either

(2.18)

is valid or

(2.19)

We use this expression in (2.17). Since , we obtain (omitting terms and )

(2.21)

or

(2.22)

Due to (2.18) we have

(2.23)

Integrating this inequality over the interval , we get

(2.24)

We substitute this expression into inequality (2.17). Since , we obtain (omitting terms and )

(2.26)

or

(2.27)

Since (2.19) holds, we have

(2.28)

Integrating this inequality over the interval , we get

(2.29)

Combining inequalities (2.24), (2.29), we conclude that, in both cases (2.18), (2.19), we have

(2.30)

and, obviously (see (1.9)),

(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

(2.32)

or (because for nonnegative and )

(2.33)

The last inequality implies

(2.34)

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

Theorem 2.6.

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

(2.35)

where

(2.36)

holds on .

Proof.

Let . Then the exponential stability of the zero solution in the metric is proved in Theorem 2.5. Now we will show that the zero solution is exponentially stable in the metric as well. As follows from Lemma 2.3, for derivative , the inequality

(2.37)

holds if We estimate and using (2.12) and inequality . We obtain

(2.38)

Since

(2.39)

inequality (2.38) yields

(2.40)

Because , we can estimate

(2.41)

Then

(2.42)

Now we get from (2.40)

(2.43)

Since

(2.44)

the last inequality implies

(2.45)

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 .

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 gives . For (1.10) the estimate

(3.1)

holds. We define an auxiliary matrix

(3.2)

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.

where .

1. (B)
   Let the matrix be nonsingular and . Let all the assumptions of part (A) with and be true. Then the derivative of the solution of problem (1.1), (1.2) satisfies on the inequality

   

   (3.4)
    

where is defined by (2.36).

Proof.

Let . We compute the full derivative of the functional (1.10) along the solutions of (1.1). For , we substitute its value from (1.1). Finally we get

(3.5)

Since the matrix is positive definite, we have

(3.6)

Now we will study the two possible cases: either

(3.7)

is valid or

(3.8)

We use this inequality in (3.6). We obtain

(3.10)

From inequality (3.7) we get

(3.11)

Integrating this inequality over the interval , we get

(3.12)

We use this inequality in (3.6) again. Since , we get

(3.14)

Because the inequality (3.8) holds, we have

(3.15)

Integrating this inequality over the interval , we get

(3.16)

Combining inequalities (3.12), (3.16), we conclude that, in both cases (3.7), (3.8), we have

(3.17)

From this, it follows

(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

(3.19)

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 .

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 , ,

(4.1)

that is, the system

(4.2)

with initial conditions (1.2). Set and

(4.3)

For the eigenvalues of matrices , , and , we get , , , , and . The matrix takes the form

(4.4)

and . Because all the eigenvalues are positive, matrix 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 . Further we have

(4.5)

Since , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.2), (35) is asymptotically stable in the metric . Finally, from (2.12) and (2.35) follows that the inequalities

(4.6)

hold on .

Example 4.2.

We will investigate system (1.1) where , ,

(4.7)

that is, the system

(4.8)

with initial conditions (1.2). Set and

(4.9)

For the eigenvalues of matrices , , and , we get , ,    , and . The matrix takes the form

(4.10)

and . Because all eigenvalues are positive, matrix 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 . Further we have

(4.11)

Since , all conditions of Theorem 2.6 are satisfied and, consequently, the zero solution of (4.8) is asymptotically stable in the metric . Finally, from (2.12) and (2.35) follows that the inequalities

(4.12)

hold on .

Remark 4.3.

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

(4.13)

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

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

(4.14)

where

(4.15)

It is easy to verify that for and , and for the equation

(4.16)

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 .

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.