# Stabilization of third-order differential equation by delay distributed feedback control

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## Abstract

There are almost no results in mathematical literature on the exponential stability of third-order delay differential equations. One of the main purposes of the paper is to fill this gap. We propose an approach to the study of stability for third-order delay differential equations.

On the basis of these results, new possibilities of stabilization by delay feedback input control are proposed.

## Keywords

Exponential stability Stabilization Delay differential equations Cauchy function W-transform## 1 Introduction

*n*th-order delay differential equations, which are quasipolynomials in the case of delay equations, were obtained in the well known books [21, 22]. In this paper we propose an absolutely different approach to the study of the exponential stability of third-order delay differential equations. Our approach is based on the idea of Azbelev’s

*W*-transform presented in the book [23] (see Chapter 5) and developed then in [13].

*I*,

*K*,

*h*are corresponding constants, \(I>0\) and \(K>0\), can describe the ship dynamics. Here \(x(t)\) is the ship deviation angle and \(\psi (t)\) is the turning angle of the rudder. Following [22] (see p. 4), we can make the following steps. Assume that the change of rudder angle \(\psi (t)\) is governed by the automatic helmsman rule

*α*,

*β*are the helmsman parameters. In practice, we can assume that \(y(t)=x(t-\tau )\). Using the representation of the general solution of Eq. (1.3)

*α*and

*β*to guarantee the exponential stability of Eq. (1.5).

The paper consists of the following sections. In Sect. 2, we formulated known results which are used in the proofs. In Sect. 3, auxiliary results on the Cauchy function for ordinary differential equations of the third order are obtained. In Sect. 4, the main results about stability of third-order delay differential equations are formulated. In Sect. 5, we prove the main theorem about stability. Conclusion, discussion of results and open problems are presented in Sect. 6.

## 2 Preliminaries

### Definition 2.1

*γ*and

*N*such that

*γ*and

*N*do not depend on \(t_{0}\geq 0\) and

*φ*,

*ψ*,

*η*.

### Definition 2.2

*t*for every fixed \(s\in [0,\infty )\) the equation

## 3 Cauchy function of an autonomous third-order ordinary differential equation

*A*,

*B*and

*C*are constants according to Definition 2.2. Its characteristic equation is

In every of these cases, the Cauchy function \(W(t,s)\) of Eq. (3.1) could be constructed according to Definition 2.2. Actually, we can solve the third-order autonomous ordinary differential Eq. (3.1) with the initial conditions \(x(s)=0\), \(x^{\prime }(s)=0\), \(x^{\prime \prime }(s)=1\). Taking this for every one of the cases (1)–(4), we obtain Lemmas 3.1–3.4 below.

Let us start with the case (1) of three different real roots.

### Lemma 3.1

*Let condition*(3.3)

*be fulfilled*,

*then*,

*in the case of*(1)

*in*(3.4),

*the Cauchy function of Eq*. (3.1)

*is of the form*

*where*

### Example 3.1

*Q*is of the form

*t*are the following:

*s*, we can obtain the inequalities

Consider now the case (2) in (3.4) of two multiple roots.

### Lemma 3.2

*Let condition*(3.3)

*be fulfilled*,

*then*,

*in the case of*(2)

*in Eq*. (3.4),

*the Cauchy function of Eq*. (3.1)

*is of the form*

*where*

Consider now the case (3) in (3.4) of three multiple roots \(k_{1}=k_{2}=k_{3}\).

### Lemma 3.3

*Let condition*(3.3)

*be fulfilled*,

*then*,

*in the case of*(3)

*in*(3.4),

*the Cauchy function of Eq*. (3.1)

*is of the form*

*and*

Consider now the case (4) in (3.4) of one real root \(k_{1}\) and two complex roots \(k_{2}=\alpha +i\beta \), \(k_{3}=\alpha -i\beta \), where we suppose below that \(\beta >0\) without loss of generality.

### Lemma 3.4

*Let condition*(3.3)

*be fulfilled*,

*then*,

*in the case of*(4)

*in*(3.4),

*the Cauchy function of Eq*. (3.1)

*is of the form*

*where*

*In this case*

*and*

## 4 Stability of third-order delay equations

It is clear that the choice of the parameters \(w_{0}\), \(w_{1}\), \(w_{2}\) and \(w_{3} \) depends on the case (1), (2), (3) and (4) in which the “constant parts” of the coefficients *A*, *B* and *C* of the given Eq. (4.2) are defined by (4.3).

### Theorem 4.1

*If the Hurwitz condition* (3.3) *for**A*, *B*, *C**defined by* (4.3) *is fulfilled and**q*, *defined by Eq*. (4.4), *satisfies the inequality*\(q<1\), *then Eq*. (4.5) *is exponentially stable*.

### Remark 4.1

We obtain the following fact.

### Corollary 4.1

*If the Hurwitz condition* (3.3) *for**A*, *B*, *C**defined by* (4.3) *is fulfilled*, *the delays*\(\tau _{ij}^{\ast }\)*and*\(\Delta a_{2j}^{\ast }\), \(\Delta b_{1j}^{\ast }\), \(\Delta c_{0j}^{\ast }\)*for*\(j=1,\ldots,m\), \(i=0,1,2\), *are sufficiently small*, *then Eq*. (4.5) *is exponentially stable*.

### Example 4.1

### Example 4.2

Denoting \(X=\Delta a^{\ast }\), \(Y=\Delta b^{\ast }\), \(Z=\Delta c^{\ast }\), we obtain a simple geometrical interpretation of this result: Eq. (4.9) under condition (4.10) is exponentially stable if the point \(M(\Delta a(t),\Delta b(t),\Delta c(t))\) for every \(t\geq 0\) is inside the pyramid formed by the planes \(X=0\), \(Y=0\), \(Z=0\) and \(\frac{X}{\frac{1}{16}}+\frac{Y}{\frac{1}{8}}+\frac{Z}{\frac{3}{14}}=1\). The last plane can be constructed as one having the intersections with the axes at the points \(( \frac{1}{16},0,0 ) \), \(( 0,\frac{1}{8},0 ) \) and \(( 0,0,\frac{3}{14} ) \).

## 5 Proofs

### Proof of Theorem 4.1

*W*-transform [23],

The condition \(q<1\), where *q* is defined by Eq. (4.4), implies that the norm \(\Vert K \Vert \) of the operator \(K:L_{\infty }\rightarrow L_{\infty }\) is less than one and this guarantees the action and boundedness of the operator \((I-K)^{-1}=I-K-K^{2}+K^{3}+\cdots\) from \(L_{\infty }\) to \(L_{\infty }\). It is clear now that, for every bounded right-hand side *f*, the solution *z* of Eq. (5.8) is bounded. From the Hurwitz condition (3.3) on Eq. (3.1) it follows that the solution \(x(t)\) and its derivatives \(x^{\prime }(t)\) and \(x^{\prime \prime }(t)\) defined by formulas (5.3) and (5.6) are bounded on the semiaxis \(t\in [0,\infty ) \) for any bounded right-hand side *f*. The Bohl–Perron theorem formulated in Lemma 2.1 (see also [23], p. 93 or [1], p. 500 in a more general formulation) claims that boundedness of solutions of Eq. (4.2) for all bounded right-hand sides *f* is equivalent to the exponential stability of Eq. (4.5). Thus the reference to the Bohl–Perron theorem completes this part of the proof.

If we do not assume that \(t-\tau _{ij}(t)\geq 0\) for \(i=0,1,2\), \(j=1,\ldots,m\), \(t\geq 0\), we can extend the coefficients on the interval \([-\tau ,0)\), where \(\tau =\operatorname{esssup}_{t\geq 0}\tau _{ij}(t)\), as follows: \(\tau _{ij}(t)\equiv 0, p_{2j}(t)\equiv \sum_{j=1}^{m}a_{2j},p_{1j}(t)\equiv \sum_{j=1}^{m}b_{1j}\) and \(p_{1j}(t)\equiv \sum_{j=1}^{m}c_{0j}\) and consider Eq. (4.1) on the interval \([-\tau ,\infty )\). Passing now to Eqs. (4.2) and (4.5) on this interval \([-\tau ,\infty )\), we can repeat the whole proof. This remark completes the proof of Theorem 4.1. □

## 6 Conclusion, discussion and some topics for future research

Note that a similar idea for stability studies of the second-order delay differential equations was proposed first in [26], developed then in [27] and the exact estimates of the integrals of the Cauchy functions (i.e. of \(w_{0}\), \(w_{1}\), \(w_{2}\)) for second-order equations were obtained in [28].

*p*and

*τ*(see [29] Chapter III, Section 16, pp. 105–106). In [30] it was proven that all solutions of the equation \(x^{\prime \prime }(t)+px(t-\tau (t))=0\) with every positive constant

*p*and nonnegative \(\tau (t)\) are bounded if and only if \(\int ^{\infty }\tau (t)\,dt<\infty \). It was considered impossible to obtain exponential stability of second-order delay equations without damping terms for the delay satisfying the inequality \(\tau (t)>\varepsilon \) for every positive

*ε*. Using an analysis of the roots of the characteristic equations, first results on the stability of the equation \(x^{\prime \prime }(t)+ax(t)-bx(t-\tau )=0\) (

*a*,

*b*and

*τ*are constant parameters) were obtained in [2, 4, 21]. In the case of variable coefficients and delays, results on the exponential stability of second-order delay equation

It is interesting to develop the method proposed in our paper for stability studies of systems of delay equations. Another possible development is to apply our “linear” results to the stability of nonlinear delay differential equations and to obtain, for example, analogous results to the ones obtained in [11, 17, 19].

## Notes

### Acknowledgements

This paper is a part of BSc and Master thesis of Shirel Shemesh and Ester Yakovi. They thank Ariel University for the possibility to unite the studies of these two degrees. This paper is a part of BSc final project of Alexander Sitkin. He thanks the Ministry of Absorption and Integration of the State of Israel for programs of new immigrants’ support.

### Authors’ information

All authors are from Department of Mathematics, Ariel University, Ariel, Israel. Professor Alexander Domoshnitsky and Dr. Roman Yavich are the staff members of this Department, Shirel Shemesh and Ester Yakovi are students of Master Degree, Alexander Sitkin is a student of BSc.

### Authors’ contributions

All authors worked and obtained the results together. All authors read and approved the final manuscript.

### Funding

Alexander Sitkin was supported by the Ministry of Absorption and Integration of the State of Israel.

### Competing interests

The authors declare that they have no competing interests.

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