Multiplicity-induced-dominancy for delay-differential equations of retarded type
Introduction
This paper addresses the asymptotic behavior of the generic delay differential equation where the unknown function y is real-valued, n is a positive integer, for are constant coefficients, and is a delay. Equation (1.1) is a delay differential equation of retarded type since the derivative of highest order appears only in the non-delayed term (see, e.g., [25] and references therein).
Systems and equations with time-delays have found numerous applications in a wide range of scientific and technological domains, such as in biology, chemistry, economics, physics, or engineering, in which time-delays are often used as simplified models for finite-speed propagation of mass, energy, or information. Due to their applications and the challenging mathematical problems arising in their analysis, they have been extensively studied in the scientific literature by researchers from several fields, in particular since the 1950s and 1960s. We refer to [4], [18], [23], [25], [30], [33], [36], [46], [48] for more details on time-delay systems and their applications.
Among others, some motivation for considering (1.1) comes from the study of linear control systems with a delayed feedback under the form where u is the control input, typically chosen in such a way that (1.2) behaves in some prescribed manner. In the absence of the delay τ and if the function y and its first derivatives are instantaneously available for measure, an usual choice is , in which case (1.2) becomes and hence, by a suitable choice of the coefficients , one may choose the roots of the characteristic equation of (1.3) and hence the asymptotic behavior of its solutions. Such a method, called pole placement, does not hold if the model includes a delay. Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions and to execute these decisions [46]. Equation (1.1) can be seen as the counterpart of (1.3) in the presence of the delay τ.
The stability analysis of time-delay systems has attracted much research effort and is an active field [2], [16], [17], [23], [26], [27], [36], [46]. One of its main difficulties is that, contrarily to the delay-free situation where Routh–Hurwitz stability criterion is available [3], there is no simple known criterion for determining the asymptotic stability of a general linear time-delay system based only on its coefficients and delays. The investigation of conditions on coefficients and delays guaranteeing asymptotic stability of solutions is a question of ongoing interest, see for instance [23], [32].
In the absence of delays, stability of linear systems and equations such as (1.3) can be addressed by spectral methods by considering the corresponding characteristic polynomial, whose complex roots determine the asymptotic behavior of solutions of the system. For systems with delays, spectral methods can also be used to understand the asymptotic behavior of solutions by considering the roots of some characteristic function (see, e.g., [4], [15], [19], [25], [36], [46], [48], [53]) which, for (1.1), is the function defined for by More precisely, the exponential behavior of solutions of (1.1) is given by the real number , called the spectral abscissa of Δ, in the sense that, for every , there exists such that, for every solution y of (1.1), one has [25, Chapter 1, Theorem 6.2]. Moreover, all solutions of (1.1) converge exponentially to 0 if and only if . An important difficulty in the analysis of the asymptotic behavior of (1.1) is that, contrarily to the situation for (1.3), the corresponding characteristic function Δ has infinitely many roots.
The function Δ is a particular case of a quasipolynomial. Quasipolynomials have been extensively studied due to their importance in the spectral analysis of time-delay systems [9], [22], [29], [37], [43], [47], [51]. The precise definition of a quasipolynomial is recalled in Section 2.1, in which we also provide some useful classical properties of this class of functions, including the fact that the multiplicity of a root of a quasipolynomial is bounded by some integer, called the degree of the quasipolynomial, which corresponds to the number of its free coefficients. In particular, according to Definition 2.2, the degree of Δ is 2n. Recent works such as [8], [9] have provided characterizations of multiple roots of quasipolynomials using approaches based on Birkhoff and Vandermonde matrices, which we briefly present in Section 2.2.
The spectral abscissa of Δ is related to the notion of dominant roots, i.e., roots with the largest real part (see Definition 2.5). Generally speaking, dominant roots may not exist for a given function of a complex variable, but they always exist for functions of the form (1.4), as a consequence, for instance, of the fact that Δ has finitely many roots on any vertical strip in the complex plane [25, Chapter 1, Lemma 4.1]. Notice also that exponential stability of (1.1) is equivalent to the dominant root of Δ having negative real part.
It turns out that, for characteristic quasipolynomials of some time-delay systems, real roots of maximal multiplicity are often dominant, a property known as multiplicity-induced-dominancy (MID for short). This link between maximal multiplicity and dominance has been suggested in [40, Chapter III, § 10; Chapter IV, § 6; Chapter V, § 7] after the study of some simple, low-order cases, but without any attempt to address the general case. Up to the authors' knowledge, very few works have considered this question in more details until recently in works such as [10], [11], [12]. MID has been shown to hold, for instance, in the case , proving dominance by introducing a factorization of Δ in terms of an integral expression when it admits a root of multiplicity 2 [12]; in the case and , using also the same factorization technique [11]; and in the case and , using Cauchy's argument principle to prove dominance of the multiple root [10].
Another motivation for studying roots of high multiplicity is that, for delay-free systems, if the spectral abscissa admits a minimizer among a class of polynomials with an affine constraint on their coefficients, then one such minimizer is a polynomial with a single root of maximal multiplicity (see [6], [14]). Similar properties have also been obtained for some time-delay systems in [35], [42], [50]. Hence, the interest in investigating multiple roots does not rely on the multiplicity itself, but rather on its connection with dominance of this root and the corresponding consequences for stability.
The aim of this paper is to extend previous results on multiplicity-induced-dominancy for low-order single-delay systems from [10], [11], [12] to the general setting of linear single-delay differential equations of arbitrary order (1.1) by exploiting the integral factorization introduced in [12]. Our main result, Theorem 3.1, states that, given any , there exists a unique choice of such that is a root of multiplicity 2n of Δ, and that, under this choice, is a strictly dominant root of Δ, determining thus the asymptotic behavior of solutions of (1.1).
The strategy of our proof of Theorem 3.1 starts by a suitable classical change of variables allowing to treat only the case and (see, for instance, [10]). The coefficients ensuring that 0 is a root of multiplicity 2n of Δ are characterized as solutions of a linear system, which allows to prove their existence as well as uniqueness (note that this characterization can be seen as a particular case of that of [8]). The key part of the proof, concerning the dominance of the multiple root at 0, makes use of a suitable factorization of Δ in terms of an integral expression that turns out to be a particular confluent hypergeometric function, whose roots have been studied in [54].
The paper is organized as follows. Section 2 presents some preliminary material on quasipolynomials, functional Birkhoff matrices, confluent hypergeometric functions, and binomial coefficients that shall be of use in the sequel of the paper. The main result of the paper is stated in Section 3, which also contains some of its consequences. In order to improve the organization and the readability of the paper and since the main ideas of the proof are themselves of independent interest, we expose the detailed proof of the main result in a dedicated section, Section 4. Section 5 provides some further remarks on the factorization of the characteristic quasipolynomial and an interpretation in terms of Laplace transforms. Finally, Section 6 illustrates the applicability of the main result and presents the P3δ toolbox, a Python software covering the numerical issues related to the proposed problem.
Section snippets
Preliminaries and prerequisites
This section contains some preliminary results on quasipolynomials (Section 2.1), functional Birkhoff matrices (Section 2.2), confluent hypergeometric functions (Section 2.3), and binomial coefficients (Section 2.4) which are used in the sequel of the paper. Before turning to the core of this section, we present the following result on the integral of the product of a polynomial and an exponential, which is rather simple but of crucial importance to prove our main result.
Proposition 2.1 Let and p be a
Statement of the main result
The main result we prove in this paper is the following characterization of real roots of maximal multiplicity of Δ and their dominance and the corresponding consequences for the stability of the trivial solution of (1.1).
Theorem 3.1 Consider the quasipolynomial Δ given by (1.4) and let . The number is a root of multiplicity 2n of Δ if and only if, for every , If (3.1) is
Proof of the main result
The proof of Theorem 3.1 consists in three steps: the normalization of the quasipolynomial Δ, the establishment of the necessary and sufficient conditions guaranteeing the maximal multiplicity, and the proof of dominance of the multiple root with respect to the remaining spectrum.
Further remarks on the factorization of the characteristic quasipolynomial
We have used in Section 4.3 the factorization (4.14) of the quasipolynomial . Recalling the relation (4.1) between and Δ, one can rewrite the factorization (4.14) in terms of the characteristic quasipolynomial Δ of (1.1) as
The proof of the factorization (4.14) in Lemma 4.5 is based on Proposition 2.1, which can be interpreted in terms of Laplace transforms as the computation of the Laplace transform of the function . In this
Improving the decay rate of a stable second-order control system without instantaneous velocity observation
Consider a stable second-order control system written under the form where are positive real numbers and is a control input. Under no control, the characteristic polynomial of this equation is , whose roots are if or if . Hence, the spectral abscissa of Δ is given by In particular, and the system is exponentially stable.
A classical problem in control theory is to
Concluding remarks
In this paper we further investigated the recently emphasized property called multiplicity-induced-dominancy for single-delay retarded delay-differential equations. Thanks to the reduction of the corresponding characteristic function into an integral representation, we have shown that characteristic spectral values of maximal multiplicity are necessarily dominant for retarded delay-differential equations of arbitrary order. Finally, to demonstrate the applicability of such a property,
Acknowledgements
This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissement d'Avenir” program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.
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2022, Bulletin des Sciences MathematiquesCitation Excerpt :If the application of hypergeometric functions to the qualitative analysis of some classes of PDEs is well-known (see, for instance, [2, Chapter I, Section 4] for the case of wave equations), it has been emphasized in [7] that the distribution of zeros of such degenerate hypergeometric functions are closely related to the spectrum location of linear functional differential equations of retarded type. More precisely, it has been shown in [7] that an exponential polynomial characterizing a linear delay differential equation with a spectral value of maximal multiplicity [1] can be factorized in terms of a Kummer hypergeometric function with positive integers as indices. The contribution of this note is threefold: first, we provide a simple counterexample of a result proposed by G. E. Tsvetkov in [16] on the location of non-asymptotic zeros of Whittaker functions.