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Stabilization of third-order differential equation by delay distributed feedback control
- Alexander Domoshnitsky^{1}Email author,
- Shirel Shemesh^{1},
- Alexander Sitkin^{1},
- Ester Yakovi^{1} and
- Roman Yavich^{1}
https://doi.org/10.1186/s13660-018-1930-5
© The Author(s) 2018
- Received: 28 August 2018
- Accepted: 2 December 2018
- Published: 12 December 2018
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
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
Definition 2.2
3 Cauchy function of an autonomous third-order ordinary differential equation
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
Example 3.1
Consider now the case (2) in (3.4) of two multiple roots.
Lemma 3.2
Consider now the case (3) in (3.4) of three multiple roots \(k_{1}=k_{2}=k_{3}\).
Lemma 3.3
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
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
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].
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].
Declarations
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.
Funding
Alexander Sitkin was supported by the Ministry of Absorption and Integration of the State of Israel.
Authors’ contributions
All authors worked and obtained the results together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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