The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions

Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu

doi:10.5802/crmath.293

Systèmes dynamiques, Théorie du contrôle

The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions

Islam Boussaada ^1,² ; Guilherme Mazanti ² ; Silviu-Iulian Niculescu ²

¹ Institut Polytechnique des Sciences Avancées (IPSA), 63 boulevard de Brandebourg, 94200 Ivry-sur-Seine, France
² Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 349-369.

Résumé

In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions.

Reçu le : 2021-07-23
Révisé le : 2021-10-25
Accepté le : 2021-10-26
Publié le : 2022-04-26

DOI : 10.5802/crmath.293

Classification : 34K35, 34K20, 93D15, 33C15, 33C90

Affiliations des auteurs :

Islam Boussaada ^{1, 2} ; Guilherme Mazanti ² ; Silviu-Iulian Niculescu ²

¹ Institut Polytechnique des Sciences Avancées (IPSA), 63 boulevard de Brandebourg, 94200 Ivry-sur-Seine, France
² Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Islam Boussaada and Guilherme Mazanti and Silviu-Iulian Niculescu},
     title = {The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: {When} delay-systems characteristics meet the zeros of {Kummer} functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {349--369},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.293},
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Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu. The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 349-369. doi : 10.5802/crmath.293. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.293/

Bibliographie
Cité par

[1] Jürgen Ackermann Der Entwurf linearer Regelungssysteme im Zustandsraum, at-Automatisierungstechnik, Volume 20 (1972) no. 1-12, pp. 297-300 | DOI | Zbl

[2] Souad Amrane; Fazia Bedouhene; Islam Boussaada; Silviu-Iulian Niculescu On qualitative properties of low-degree quasipolynomials: further remarks on the spectral abscissa and rightmost-roots assignment, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., Volume 61(109) (2018) no. 4, pp. 361-381 | MR | Zbl

[3] Fatihcan M. Atay Balancing the inverted pendulum using position feedback, Appl. Math. Lett., Volume 12 (1999) no. 5, pp. 51-56 | DOI | MR | Zbl

[4] Tamas Balogh; Islam Boussaada; Tamas Insperger; Silviu-Iulian Niculescu Towards an MID-based Delayed Design for Arbitrary-order Dynamical Systems with a Mechanical Application, IFAC-PapersOnLine, Volume 53 (2020) no. 2, pp. 4375-4380 (Proceedings of the 21th IFAC World Congress) | DOI

[5] Tamas Balogh; Islam Boussaada; Tamas Insperger; Silviu-Iulian Niculescu Conditions for stabilizability of time-delay systems with real-rooted plant, International Journal of Robust and Nonlinear Control, Volume 32 (2022) no. 6, pp. 3206-3224 (Special Issue:System Theory and Delay: In honour of Vladimir Kharitonov) | DOI

[6] Roberto Barrio; Juan Manuel Peña Basis conversions among univariate polynomial representations, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 4, pp. 293-298 | DOI | MR | Zbl

[7] Fazia Bedouhene; Islam Boussaada; Silviu-Iulian Niculescu Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 9-10, pp. 1011-1032 | MR | Zbl

[8] Richard Bellman; Kenneth L. Cooke Differential-difference equations, Academic Press Inc., 1963, xvi+462 pages

[9] Amina Benarab; Islam Boussaada; Karim Trabelsi; Guilherme Mazanti; Catherine Bonnet The MID property for a second-order neutral time-delay differential equation, 2020 24th International Conference on System Theory, Control and Computing (2020), pp. 202-207

[10] Carlos A. Berenstein; Roger Gay Complex analysis and special topics in harmonic analysis, Springer, 1995, x+482 pages | DOI | MR

[11] Serge Bernstein Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Charkow Ges. (2), Volume 13 (1912) no. 1, pp. 1-2 | Zbl

[12] Vincent D. Blondel; Mert Gürbüzbalaban; Alexandre Megretski; Michael L. Overton Explicit solutions for root optimization of a polynomial family with one affine constraint, IEEE Trans. Autom. Control, Volume 57 (2012) no. 12, pp. 3078-3089 | DOI | MR | Zbl

[13] Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions, Bull. Sci. Math., Volume 174 (2022), 103093, 12 pages | MR | Zbl

[14] Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu; Julien Huynh; Franck Sim; Matthieu Thomas Partial pole placement via delay action: A Python software for delayed feedback stabilizing design, 2020 24th International Conference on System Theory, Control and Computing (2020), pp. 196-201 | DOI

[15] Islam Boussaada; Silviu-Iulian Niculescu Characterizing the codimension of zero singularities for time-delay systems: a link with Vandermonde and Birkhoff incidence matrices, Acta Appl. Math., Volume 145 (2016), pp. 47-88 | DOI | MR | Zbl

[16] Islam Boussaada; Silviu-Iulian Niculescu Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: a Vandermonde-based approach, IEEE Trans. Autom. Control, Volume 61 (2016) no. 6, pp. 1601-1606 | DOI | MR | Zbl

[17] Islam Boussaada; Silviu-Iulian Niculescu; Ali El Ati; Redamy Pérez-Ramos; Karim Trabelsi Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design, ESAIM, Control Optim. Calc. Var. (2020), 57, 34 pages | MR | Zbl

[18] Islam Boussaada; Sami Tliba; Silviu-Iulianand Niculescu; Hakki Ulaş Ünal; Tomáš Vyhlídal Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system, Linear Algebra Appl., Volume 542 (2018), pp. 589-604 | DOI | MR

[19] Islam Boussaada; Hakki Ulaş Ünal; Silviu-Iulian Niculescu Multiplicity and Stable Varieties of Time-Delay Systems: A Missing Link, Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS) (2016), pp. 188-194

[20] David Brethé; Jean Jacques Loiseau An effective algorithm for finite spectrum assignment of single-input systems with delays, Math. Comput. Simul., Volume 45 (1998) no. 3, pp. 339-348 | DOI | MR | Zbl

[21] Herbert Buchholz The confluent hypergeometric function with special emphasis on its applications, Springer Tracts in Natural Philosophy, 15, Springer, 1969, xviii+238 pages (translated from the German by H. Lichtblau and K. Wetzel) | DOI | MR

[22] Raymond Chen Output feedback stabilization of linear systems, Ph. D. Thesis, University of Florida (1979)

[23] Kenneth L. Cooke; Pauline van den Driessche On zeroes of some transcendental equations, Funkc. Ekvacioj, Volume 29 (1986) no. 1, pp. 77-90 | MR | Zbl

[24] Jean Michel Coron; Simona Oana Tamasoiu Feedback stabilization for a scalar conservation law with PID boundary control, Chin. Ann. Math., Ser. B, Volume 36 (2015) no. 5, pp. 763-776 | DOI | MR | Zbl

[25] Driss Drissi Characterization of Kummer hypergeometric Bernoulli polynomials and applications, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 10, pp. 743-751 | DOI | MR | Zbl

[26] Koen Engelborghs; Michel Dambrine; Dirk Roose Limitations of a class of stabilization methods for delay systems, IEEE Trans. Autom. Control, Volume 46 (2001) no. 2, pp. 336-339 | DOI | MR | Zbl

[27] Arthur Erdélyi; Wilhelm Magnus; Fritz Oberhettinger; Francesco G. Tricomi Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., 1981, xiii+302 pages (based on notes left by Harry Bateman, with a preface by Mina Rees and a foreword by E. C. Watson, reprint of the 1953 original) | MR

[28] Jack K. Hale; Sjoerd M. Verduyn Lunel Introduction to functional differential equations, Applied Mathematical Sciences, 99, Springer, 1993 | DOI

[29] Godfrey H. Hardy On the Zeroes of Certain Classes of Integral Taylor Series. Part II.–On The Integral Function $\sum_{n = 0}^{\infty} \frac{x^{n}}{(n + a) 8^{n}!}$ and Other Similar Functions, Proc. Lond. Math. Soc., Volume 2 (1905), pp. 401-431 | DOI | MR

[30] N. D. Hayes Roots of the Transcendental Equation Associated with a Certain Difference-Differential Equation, J. Lond. Math. Soc., Volume s1-25 (1950) no. 3, pp. 226-232 | DOI | MR | Zbl

[31] Einar Hille Oscillation theorems in the complex domain, Trans. Am. Math. Soc., Volume 23 (1922) no. 4, pp. 350-385 | DOI | MR

[32] Allan M. Krall The Root Locus Method: A Survey, SIAM Rev., Volume 12 (1970) no. 1, pp. 64-72 | DOI | MR | Zbl

[33] Dan Ma; Islam Boussaada; Catherine Bonnet; Silviu-Iulian Niculescu; Jie Chen Multiplicity-Induced-Dominancy extended to neutral delay equations: Towards a systematic PID tuning based on Rightmost root assignment, ACC 2020 - American Control Conference (2020)

[34] Andrzej Z. Manitius; Andrzej W. Olbrot Finite spectrum assignment problem for systems with delays, IEEE Trans. Autom. Control, Volume 24 (1979) no. 4, pp. 541-552 | DOI | MR | Zbl

[35] Guilherme Mazanti; Islam Boussaada; Silviu-Iulian Niculescu Multiplicity-induced-dominancy for delay-differential equations of retarded type, J. Differ. Equations, Volume 286 (2021), pp. 84-118 | DOI | MR | Zbl

[36] Guilherme Mazanti; Islam Boussaada; Silviu-Iulian Niculescu; Yacine Chitour Effects of Roots of Maximal Multiplicity on the Stability of Some Classes of Delay Differential-Algebraic Systems: The Lossless Propagation Case, Proceeding of 24th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2021) (IFAC-PapersOnLine) (2021)

[37] Guilherme Mazanti; Islam Boussaada; Silviu-Iulian Niculescu; Tomáš Vyhlídal Spectral dominance of complex roots for single-delay linear equations, IFAC 2020 - 21st IFAC World Congress (IFAC-PapersOnLine) (2020)

[38] Wim Michiels; Koen Engelborghs; P. Vansevenant; Dirk Roose Continuous pole placement for delay equations, Automatica, Volume 38 (2002) no. 5, pp. 747-761 | DOI | MR | Zbl

[39] Wim Michiels; Silviu-Iulian Niculescu Stability, control, and computation for time-delay systems: An eigenvalue-based approach, Advances in Design and Control, 27, Society for Industrial and Applied Mathematics, 2014, xxiv+435 pages | DOI | MR

[40] Wim Michiels; Tomas Vyhlidal An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type, Automatica, Volume 41 (2005) no. 6, pp. 991-998 | DOI | MR | Zbl

[41] Yu. I. Neĭmark The structure of the $D$ -decomposition of the space of quasipolynomials and the diagrams of Vyšnegradskiĭ and Nyquist, Dokl. Akad. Nauk SSSR, n. Ser., Volume 60 (1948), pp. 1503-1506 | MR

[42] Nikola Obreschkoff Nullstellen linearer Kombinationen von Exponentialfunktionen, Jber. der Deutsch. Math. Verein., Volume 37 (1928), pp. 81-84

[43] Andrzej W. Olbrot Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays, IEEE Trans. Autom. Control, Volume 23 (1978) no. 5, pp. 887-890 | DOI | MR | Zbl

[44] NIST handbook of mathematical functions (Frank W. J. Olver; Daniel W. Lozier; RonaldF. Boisvert; Charles W. Clark, eds.), Cambridge University Press, 2010, xvi+951 pages | MR

[45] Edmund Pinney Ordinary difference-differential equations, University of California Press, 1958, xii+262 pages | MR

[46] George Pólya Über die Nullstellen gewisser ganzer Funktionen, Math. Z., Volume 2 (1918) no. 3, pp. 352-383 | DOI | Zbl

[47] George Pólya; Gabor Szegő Problems and theorems in analysis II: Theory of functions. Zeros. Polynomials. Determinants. Number theory. Geometry, Springer, 1997

[48] George Pólya; Gabor Szegő Problems and theorems in analysis. I Series, integral calculus, theory of functions, Classics in Mathematics, Springer, 1998, xx+389 pages (translated from the German by Dorothee Aeppli, reprint of the 1978 English translation) | DOI | MR

[49] Project Jupyter; Matthias Bussonnier; Jessica Forde; Jeremy Freeman; Brian Granger; Tim Head; Chris Holdgraf; Kyle Kelley; Gladys Nalvarte; Andrew Osheroff; M Pacer; Yuvi Panda; Fernando Perez; Benjamin Ragan-Kelley; Carol Willing Binder 2.0 - Reproducible, interactive, sharable environments for science at scale, Proceedings of the 17th Python in Science Conference (2018), pp. 113-120 | DOI

[50] Yitshak M. Ram; John E. Mottershead; Maryam G. Tehrani Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra Appl., Volume 434 (2011) no. 7, pp. 1689-1696 | DOI | MR | Zbl

[51] Adrián Ramírez; Sabine Mondié; Rubén Garrido; Rifat Sipahi Design of proportional-integral-retarded (PIR) controllers for second-order LTI systems, IEEE Trans. Autom. Control, Volume 61 (2016) no. 6, pp. 1688-1693 | DOI | MR | Zbl

[52] Anatolii Mechislavovich Sedletskii On the Zeros of Laplace Transforms., Math. Notes, Volume 76 (2004), pp. 824-833 | DOI | MR

[53] Gábor Stépán Retarded dynamical systems: stability and characteristic functions, Pitman Research Notes in Mathematics Series, 210, Longman Scientific & Technical; John Wiley & Sons, 1989, viii+151 pages | MR

[54] Edward C. Titchmarsh The zeros of certain integral functions, Proc. Lond. Math. Soc., Volume 2 (1926) no. 1, pp. 283-302 | DOI | MR | Zbl

[55] Tomas Vyhlidal; Wim Michiels; Pavel Zitek Quasi-direct pole placement for time delay systems applied to a heat transfer set-up, IFAC Proceedings Volumes, Volume 42 (2009) no. 14, pp. 325-330 (8th IFAC Workshop on Time-Delay Systems) | DOI

[56] Qing-Gou Wang; Tong Heng Lee; Kok Kiong Tan Finite spectrum assignment for time-delay systems, Lecture Notes in Control and Information Sciences (NCIS), 239, Springer, 1999 | DOI

[57] Franck Wielonsky A Rolle’s theorem for real exponential polynomials in the complex domain, J. Math. Pures Appl., Volume 80 (2001) no. 4, pp. 389-408 | DOI | MR | Zbl

[58] Edward M. Wright Stability criteria and the real roots of a transcendental equation, J. Soc. Ind. Appl. Math., Volume 9 (1961), pp. 136-148 | DOI | MR | Zbl

[59] Peter Wynn On the zeros of certain confluent hypergeometric functions, Proc. Am. Math. Soc., Volume 40 (1973), pp. 173-182 | DOI | MR | Zbl

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