W-transform for exponential stability of second order delay differential equations without damping terms
- Alexander Domoshnitsky^{1}Email author,
- Abraham Maghakyan^{1} and
- Leonid Berezansky^{2}
https://doi.org/10.1186/s13660-017-1296-0
© The Author(s) 2017
Received: 16 November 2016
Accepted: 4 January 2017
Published: 17 January 2017
Abstract
In this paper a method for studying stability of the equation \(x^{\prime \prime }(t)+\sum_{i=1}^{m}a_{i}(t)x(t- \tau_{i}(t))=0\) not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation \(x^{\prime \prime}(t)+\sum_{i=1}^{m}a_{i}(t)x(t)=0\) is not exponentially stable, the delay equation can be exponentially stable.
Keywords
MSC
1 Introduction
Various applications of equation (1.1) and its generalizations can be found, for example, in the theory of self-excited oscillations, in oscillation processes in a vacuum tube, in dynamics of an auto-generator, in description of processes of in-feed grinding and cutting (see [1]); on position control in mechanical engineering, on electromechanical systems, and on combustion engines [2]. The problem of stabilizing the rolling of a ship by the activated tanks method in which ballast water is pumped from one position to another was reduced in [3] to analysis of stability of the second order delay equation.
Our method reduces the study of stability of equations without damping term to the study of stability of corresponding equations with damping term. The stability of autonomous delay differential equations of the second order with damping terms was studied in [1, 9, 10], which in the case of delay differential equations apply quasi-polynomials and not polynomials as in the case of ordinary differential equations. That is why a special approach for the analysis of the characteristic equations is used [13]. The technique of the Lyapunov functions was used in the works [14–16] and the technique of the fixed point theorems in [17]. The technique of non-oscillation and positivity of the Cauchy functions for studying stability of delay equations was proposed [18] and then developed in [19].
Let us formulate several definitions concerning stability.
Definition 1
Definition 2
It is well known that for equation (1.1) with bounded delays these two definitions are equivalent [20].
The paper is built as follows. In the first section we describe known results on asymptotic properties of second order delay equations. In Section 2, we formulate the main results of the paper and compare them with known results. Auxiliary assertions can be found in Section 3. The proofs of the assertions, formulated in Section 2, can be found in Section 4.
2 Formulation of main results
Theorem 1
Theorem 2
Theorem 3
Corollary 1
Example 1
Let us set \(a=\sqrt{5}+0.01\), \(b=\sqrt{5}\), \(\theta -\tau =0.1\). In this case we have \(b^{2}(\theta -\tau)^{2}>4(a-b)\) and condition (2.11) is fulfilled if \(\theta <1.876\).
Corollary 2
Example 2
Let us set \(a=2.01\), \(b=2\), \(\theta -\tau =0.1\). In this case we have \(b^{2}(\theta -\tau)^{2}=4(a-b)\) and condition (2.12) is fulfilled if \(\theta <4\).
Corollary 3
Example 3
Let us set \(a=2\), \(b=1.99\), \(\theta -\tau =0.1\). In this case we have \(b^{2}(\theta -\tau)^{2}<4(a-b)\) and condition (2.13) is fulfilled if \(\theta <0.23\).
3 Estimates of integrals of the Cauchy functions for auxiliary equations
Consider all possible cases: (1) \(A^{2}>4B\), (2) \(A^{2}=4B\), (3) \(A^{2}<4B\).
Lemma 1
Lemma 2
Lemma 3
The proofs of these lemmas can be found in [21], Lemmas 3.1-3.3.
4 Proofs of main theorems
The following result, which we formulate here for equation (1.1), is known as the Bohl-Perron theorem.
Lemma 4
Theorem 4
Remark 2
It is clear from inequality (4.7) that in the case, when the coefficients \(a_{i}(t)-b_{i}(t)\) and \(b_{i}(t)(\theta_{i}(t)-\tau_{i}(t))\) (\(1=1,\ldots,m\)) are close to constants, the second and fourth terms are small, and in the case of small delays \(\theta_{i}(t)\) and \(\tau_{i}(t) \) (\(1=1,\ldots,m\)), the first and the third terms are small. We can draw the conclusion that in this case equation (1.1) preserves an exponential stability of equation (2.1).
Proof
We have to prove that, for every essentially bounded function \(f(t)\), the solution \(x(t)\) is also bounded on the semiaxis \(t\in [ 0,+\infty)\). To prove exponential stability of (4.8) we assume the existence of an unbounded solution \(x(t)\) and demonstrate that this is impossible.
The inequality (4.7) implies that the norm \(\Vert K\Vert \) of the operator \(K:L_{\infty }\rightarrow L _{\infty }\) is less than one. In this case, there exists the bounded operator \((I-K)^{-1}\). For every bounded right-hand side the solution z of equation (4.18) is bounded.
In the case \(A>0\) and \(B>0\), the Cauchy function \(W(t,s)\) and its derivative \(W_{t}^{\prime }(t,s)\) satisfy exponential estimates. The boundedness of the solution x of equation (1.1) and its derivative \(x^{\prime }\) follow now from the boundedness of z.
We have got a contradiction with our assumption that the solution \(x(t)\) is unbounded on the semiaxis.
Thus solutions \(x(t)\) are bounded on the semiaxis for a bounded right-hand side \(f(t)\). Then by Lemma 4 the Cauchy function \(C(t,s)\) of equation (1.1) and the solutions \(x_{1}\) and \(x_{2}\) satisfy the exponential estimates.
This completes the proof of Theorem 4. □
To prove Theorems 1-3 we set the norms of \(\Vert W\Vert \), \(\Vert W_{t}^{\prime }\Vert \) and \(\Vert W_{tt}^{\prime \prime }\Vert \) obtained in Lemmas 1-3 into Theorem 4.
The proofs of Corollaries 1-3 are results of substitution of \(\Vert W\Vert \), \(\Vert W_{t} ^{\prime }\Vert \) and \(\Vert W_{tt}^{\prime \prime }\Vert \) into Theorems 1-3, when we take into account that \(A=b(\theta -\tau)\), \(B=a-b\), \(\Delta A=\Delta B=0\).
5 Conclusion
6 Discussion and some topics for future research
The idea of applications of W-transform for second order delay differential equations first appeared in paper [22]. Lemmas 1 and 2 were taken from [21]. All other results in the paper are new and have not been published before.
Explicit integral estimates of the fundamental function and its derivatives we obtain here only for the simplest equation: ordinary differential equation with constant coefficients. It is interesting to obtain such estimates for the ordinary differential equation with variable coefficients or for delay differential equation with constant coefficients. It will allow one to improve the results obtained in this paper.
The next problem is to apply the W-transformation method for the instability of delay differential equations of the second order.
Declarations
Acknowledgements
The authors express their sincere gratitude to the referees for careful reading of the manuscript and valuable suggestions, which helped to improve the paper.
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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