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2023 | OriginalPaper | Buchkapitel

Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications

verfasst von : Silviu-Iulian Niculescu, Xu-Guang Li, Arben Çela

Erschienen in: Controlling Delayed Dynamics

Verlag: Springer International Publishing

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Abstract

This chapter addresses the stability analysis of linear dynamical systems represented by delay differential equations with a focus on the effects induced by the delay, seen as a parameter, on the dynamical behavior. More precisely, we propose a frequency-sweeping framework for treating the problem, and the stability problem is reformulated in terms of properties of frequency-sweeping curves. The presentation is teaching-oriented and focuses more on discussing the main ideas of the method and their illustration through appropriate examples and less on explicit proofs of the results. Some applications from Life Sciences complete the presentation.

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Vorheriges Kapitel Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach

Nächstes Kapitel Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods

The delays may be constant or time-varying, distributed or not over a finite or infinite time-interval, depending on the state vector or not.

The origins of root locus go back to the works of Evans at the end of the 40s (see, e.g., Evans 1950 and the references therein).

To the best of the authors’ knowledge, the notion of “frequency-sweeping” was formally introduced by Chen and Latchman (1995), Chen (1995) into a different methodological frame: robust analysis with respect to the delay parameter, seen as an uncertainty, see also Niculescu (2001).

Under the assumption that the system free of delays is asymptotically stable, the delay margin is the maximal value \(\tau _m>0\) such that the asymptotic stability is guaranteed for all delays inside the interval \([0,\tau _m)\); see also Chen et al. (1995), Chen (1995).

In this chapter, the root loci are numerically generated by using the DDE-BIFTOOL (Engelborghs et al. 2002; Sieber et al. 2016).

For further discussions on such topics, we refer to Michiels and Niculescu (2014) and the references therein.

Such a case may occur in the case of neutral DDEs or if the coefficients of the quasipolynomials depend on the delay parameters.

For a simple and elementary proof, we refer to Shaughnessy and Kashiwagi (1969).

It simply guarantees the exponential stability of the trivial solution of the corresponding delay-difference equation.

The relative degree is defined by \(\deg (a_0)-\deg (a_1)\).

More precisely, in this configuration (i.e., improperly-posed approximation), a characteristic root appears on the real axis in \(\mathbb {C}_+\) from \(+\infty \) when the delay is increased from 0 to \(0_+\).

The common roots \(a_0\) and \(a_1\) on \({\text {i}}\mathbb {R}\) are also invariant roots w.r.t. \(\tau \).

Such a value always exists and it may be 0.

It is easy to see the way the roots of f and \(f_a\) are linked. For instance, for any pair \(({\text {i}}\omega _s,\tau _s)\in \mathbb {R}_+^*\times \mathbb {R}_+^*\) satisfying \(f({\text {i}}\omega _s,\tau _s)=0\), \(f_a(\omega _s,z_s)=0\), where \(z_s=e^{-{\text {i}}\omega _s}\), etc.

To the best of the authors’ knowledge, during the 70s, the notions of (stability) switches/reversals appear in Cooke’s publications.

I.e., the whole set for \(\tau \ge 0\) such that \(NU(\tau )=0\) excluding the possible critical points.

For an elementary introduction to Puiseux series, we refer to Casas-Alvero (2000).

More precisely, if \(\lambda _\alpha \) is a critical imaginary root for \(\tau =\tau _{\alpha ,0}\), then the system has a critical imaginary root \(\lambda _\alpha \) for all \(\tau =\tau _{\alpha ,0}+k\frac{{2\pi }}{{{\omega _\alpha }}}\), \(k \in \mathbb {N}\).

Or other software for scientific computation.

Here, \(\left( {\begin{array}{c}i+l\\ i\end{array}}\right) \) denotes the number of i-combinations from a set of \(i+l\) elements.

For instance, in our case, for each k, the Puiseux series has multiple conjugacy classes; next, for each k, the Puiseux series involves many degenerate terms, and finally, the structure of Puiseux series is variable w.r.t. different k.

Since some DDEs may have critical imaginary roots when \(\tau = 0\), one needs to consider a sufficiently “small” delay value \(\varepsilon >0\).

The critical root \(\lambda ={\text {i}}\) is a double critical imaginary root at \(\tau =\pi \).

In our case, one characteristic root is located in the right-half plane \(\mathbb {C}_+\) and the other in the left-half plane \(\mathbb {C}_-\).

Both the characteristic functions (3) and (11) are standard quasipolynomials.

The stability of the trivial solution of the neutral delay-difference equation is guaranteed. It is worth mentioning that the case \(\beta =0\) corresponds to the retarded DDE and it was addressed in the previous section.

It is worth mentioning that \(\lambda ={\text {i}}\) is simple at all \(\tau = (2k + 1)\pi \), \(k \in \mathbb {N}_+\).

In this case, \({f_i}( \cdot ) = \tanh ( \cdot )\), \(i \in \llbracket 1,4 \rrbracket \) verifying \({f_i}(0) = 0\) and \(f'_i (0) = 1\), \(i \in \llbracket 1,4 \rrbracket \).

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Titel: Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications
verfasst von: Silviu-Iulian Niculescu
Xu-Guang Li
Arben Çela
Verlag: Springer International Publishing
Buch: Controlling Delayed Dynamics
Print ISBN: 978-3-031-00981-5

Electronic ISBN: 978-3-031-01129-0

Copyright-Jahr: 2023
DOI: https://doi.org/10.1007/978-3-031-01129-0_5

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