#### Abstract

Stability of the first-order neutral delay equation with complex coefficients is studied, by analyzing the existence of stability switches.

#### 1. Introduction

Delay differential equations (DDEs), and specifically DDEs of neutral type, appear in different scientific and technical problems, as in population dynamics, the modeling of networks containing lossless transmission lines, or the study of oscillations in elastic bars (see [1–3]).

The aim of this paper is to characterize the stability of the equation where is a constant delay and , , and are complex parameters.

In [4], Cahlon and Schmidt characterized the asymptotic behavior of the zero solution of the retarded equation with complex coefficients by transforming the complex equation into two coupled real DDEs. In [5], Wei and Zhang considered the same equation and, by studying the distribution of the roots of the characteristic equation for the associated real differential system with delay, analyzed the existence of stability switches [6–8].

Transforming a complex DDE into two coupled real DDEs to analyze its stability has some drawbacks, as, in general, the orders of the characteristic quasipolynomials to be analyzed double, and, since the study of the distribution of their roots is much more complicated as their degrees increase, it becomes very difficult to obtain necessary and sufficient conditions of stability [1, 9, 10].

To avoid this problem, recently Li et al. [11] presented a method for directly analyzing the stability of complex DDEs on the basis of stability switches. Their results generalize those for real DDEs, thus greatly reducing the complexity of the analysis. In this paper, the results developed in [11] will be used to study the stability switches of the zero solution of the neutral equation (1).

#### 2. Previous Results

For the sake of clarity in the exposition, we recall some results that will be used later, that may be found in [11].

Consider time-delay systems with a characteristic equation in the following form: where and and are complex polynomial of order and respectively, with either or and , where are the highest order coefficients of and , respectively. We will also assume that which states that is not a root of (3), and that and have no roots on the imaginary axis simultaneously.

The assumed conditions are those required to use the result of Li et al. [11] stated below, but in other cases the stability of the system is mostly understood. Thus, if , or , and , there are clusterings of roots on a vertical line in the right half plane or the imaginary axis, and in the first of these cases the zero solution is unstable for all delays. Also, if (4) fails or and have common roots on the imaginary axis, these common roots, or , are roots of for all , and for all delays the zero solution is not asymptotically stable.

Consider the function If is a zero of , then there are an infinite number of delays corresponding to satisfying For these critical values, the following theorem [11, Theorem 1] characterizes the variation of the number of zeros with nonnegative real parts of , in terms of the order and sign of the first nonzero derivative of evaluated at , extending to the complex coefficients setting previous results valid only for real DDEs [8, 12, 13].

Theorem 1. *Assume that Let be the number of zeros with nonnegative real parts of , and let be an integer such that , and for all . Then,*(a)* keeps unchanged as increases along if is even,*(b)*when is odd, increases by one if and decreases by one if , as increases along .*

It should be noted that it is possible for a particular delay to produce more than one pure imaginary root, so that, in Theorem 1, the change in refers to the change resulting from the specific critical value .

#### 3. Stability Analysis of the First-Order Neutral DDE

Consider the complex DDE (1), where The characteristic equation associated with (1) is so that where Since the order of both polynomials is , we must demand that As pointed out before, if there are clusterings of roots on a vertical line in the right half plane, when , in which case the system is unstable for all delays, or on the imaginary axis, when . In this last critical case, the stability of the systems is not so clear, and a detailed, different type of analysis is required, as carried out in [8, pages 70–72] in the much simpler setting of real coefficients.

The following lemma gives , the number of zeros with nonnegative real parts of when the delay is zero.

Lemma 2. *If
**
then . Otherwise, .*

*Proof. *Consider the equation
Then
and therefore
and thus . If , there is only one root with real part nonnegative, and hence .

Now consider the function and calculate its zeros. One gets where First, we assume that and consider several subcases.

If , then has no real root, and therefore the stability of the zero solution of (1) does not change for any .

If , then has two real roots, , and , such that this last possibility being excluded from the analysis, since is a root of (3), a contradiction to (4).

Consider the case where . Substituting into (8), and separating the real and imaginary parts, one gets obtaining the following two sets of values of for which there are roots, where and where and

Since one has Therefore, according to Theorem 1, the number of the characteristic roots with nonnegative real parts increases by one as passes through and decreases by one as passes through .

If , that is, if the zero solution of (1) is stable for , then it must follow that , since cannot become negative. There are stability switches when the delays are such that Since the intervals become smaller with increasing , so that eventually, for a certain , Thus, the distribution of delays is and there is only a finite number of stability switches, with the system becoming unstable for . Under these conditions, if , that is, if the zero solution of (1) is unstable for , the system cannot be stabilized.

If , stability switches occur when the distribution of delays is in which case there are switches from instability to stability to instability. Once , stability switch stops, so that the system becomes unstable for .

The conditions on the delays for the above orderings to be valid can be formulated directly from (22) to (25).

Now we study the case when . Proceeding as before, there are two sets of critical values of delays, and , corresponding to and , respectively, but now it holds that and are both positive. Hence, the number of the characteristic roots with nonnegative real parts increases by one as passes through or . If , then there is such that the zero solution of (1) is stable for and it becomes unstable for . If , the zero solution of (1) never becomes stable for any .

To finish the analysis when , let so that has a repeated real root, For this root, as in the previous cases, there is a set of critical delays . Since , we have to consider the second derivative, According to Theorem 1, since is even, keeps unchanged as increases along . We conclude that the stability of the zero solution of (1) does not change for any .

Finally, consider the case when The analysis and results of this case are much similar to the previous one, and we will omit the details. There are only minor differences when and .

In this subcase, the function has two roots, , with corresponding sets of critical delays, with and satisfying the expressions (23) and (25), respectively. It holds that and , and hence, from Theorem 1, the number of the characteristic roots with nonnegative real parts decreases by one as passes through and increases by one as passes through . Also, since only a finite number of stability switches may exist.

If , then it must follow that , and there are switches from stability to instability when the distribution of the critical delays is Once , stability switch stops and the system becomes unstable for .

Similarly, if , stability switches may occur for the distribution of critical delays Once , stability switch stops and the system remains unstable.

In summary, the following theorem has been established.

Theorem 3. *Under the hypotheses of Theorem 1, consider the first-order complex neutral delay equation (1), where satisfies condition (11).*(a)*If , and ,(1) if , then the zero solution of (1) is stable for all ;(2)if , then the zero solution of (1) is unstable for all .*(b)

*If , , and ,(1)*(c)

*if , then there is such that the zero solution of (1) is stable for and it becomes unstable for ;*(2)*if , the zero solution of (1) is always unstable.**If , , and ,(1)*(d)

*if and the distribution of critical delays is then the zero solution of (1) is asymptotically stable for and and unstable for and ;*(2)*if and the distribution of critical delays is then the zero solution of (1) is asymptotically stable for and unstable for , and .**If , , and ,(1)*

*if and the distribution of critical delays is then the zero solution of (1) is asymptotically stable for and , and unstable for and ;*(2)*if and the distribution of critical delays is then the zero solution of (1) is asymptotically stable for and unstable for , , and .*Finally, we will consider the case for which

If , then has no real root, and therefore the stability of the zero solution of (1) does not change for any .

If , then has two real roots, with corresponding sets of critical delays where , and where , and

In this case, it holds that and are both positive, and hence, from Theorem 1, the number of the characteristic roots with nonnegative real parts increases by one as passes either through or . Thus, the following theorem has been established.

Theorem 4. *Under the hypotheses of Theorem 1, consider (1) satisfying condition (11). If ,*(a)*if , then the zero solution of (1) is always stable if and unstable if ;*(b)*if , if then there is such that the zero solution of (1) is stable for and it becomes unstable for . If , the zero solution of (1) never becomes stable for any .*

#### Acknowledgments

The authors wish to thank the reviewers for useful comments and suggestions, which improved the paper. F. Rodríguez is grateful for financial support from project FEEDBACK (CGL2011-30515-C02-01), funded by the Spanish Ministry of Science and Innovation.