Approximate and exact controllability of linear difference equations

Chitour, Yacine; Mazanti, Guilherme; Sigalotti, Mario

doi:10.5802/jep.112

Chitour, Yacine ¹; Mazanti, Guilherme ²; Sigalotti, Mario ³

¹ Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS — CentraleSupelec — Université Paris-Sud 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France
³ Inria & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions 75005 Paris, France

Journal de l’École polytechnique - Mathématiques, Volume 7 (2020), pp. 93-142.

Abstract
Résumé

In this paper, we study approximate and exact controllability of the linear difference equation $x (t) = \sum_{j = 1}^{N} A_{j} x (t - Λ_{j}) + B u (t)$ in $L^{2}$ , with $x (t) \in ℂ^{d}$ and $u (t) \in ℂ^{m}$ , using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$ , and some suitable matrix coefficients. When $Λ_{1}, \dots, Λ_{N}$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^{2}$ . The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.

Cet article traite de la contrôlabilité approchée et exacte de l’équation aux différences linéaire $x (t) = \sum_{j = 1}^{N} A_{j} x (t - Λ_{j}) + B u (t)$ dans $L^{2}$ , avec $x (t) \in ℂ^{d}$ et $u (t) \in ℂ^{m}$ , en s’appuyant sur une formule de représentation de la solution $x$ en termes de la condition initiale, du contrôle $u$ et de coefficients matriciels appropriés. Lorsque $Λ_{1}, \dots, Λ_{N}$ sont commensurables, les contrôlabilités approchée et exacte sont équivalentes et peuvent être caractérisées par un critère de type Kalman. Cet article s’attache à donner des caractérisations des contrôlabilités approchée et exacte sans hypothèse de commensurabilité. Dans le cas d’un système bi-dimensionnel avec deux retards, nous obtenons une caractérisation explicite des contrôlabilités approchée et exacte en termes des paramètres du problème. Pour le cas général, nous prouvons que la contrôlabilité approchée de zéro vers les états constants est équivalente à la contrôlabilité approchée dans $L^{2}$ . Le résultat correspondant à la contrôlabilité exacte est vrai au moins pour les systèmes bi-dimensionnels avec deux retards.

Received: 2019-02-15
Accepted: 2019-10-04
Published online: 2019-11-08

MR Zbl

DOI: 10.5802/jep.112

Classification: 39A06, 93B05, 93C65
Keywords: Linear difference equation, delay, approximate controllability, exact controllability
Mot clés : Équation aux différences linéaire, retard, contrôlabilité approchée, contrôlabilité exacte

Author's affiliations:

Chitour, Yacine ¹; Mazanti, Guilherme ²; Sigalotti, Mario ³

@article{JEP_2020__7__93_0,
     author = {Chitour, Yacine and Mazanti, Guilherme and Sigalotti, Mario},
     title = {Approximate and exact controllability of linear difference equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique - Math\'ematiques},
     pages = {93--142},
     publisher = {Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.112},
     mrnumber = {4033751},
     zbl = {07128378},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jep.112/}
}

TY  - JOUR
AU  - Chitour, Yacine
AU  - Mazanti, Guilherme
AU  - Sigalotti, Mario
TI  - Approximate and exact controllability of linear difference equations
JO  - Journal de l’École polytechnique - Mathématiques
PY  - 2020
SP  - 93
EP  - 142
VL  - 7
PB  - Ecole polytechnique
UR  - https://www.numdam.org/articles/10.5802/jep.112/
DO  - 10.5802/jep.112
LA  - en
ID  - JEP_2020__7__93_0
ER  -

%0 Journal Article
%A Chitour, Yacine
%A Mazanti, Guilherme
%A Sigalotti, Mario
%T Approximate and exact controllability of linear difference equations
%J Journal de l’École polytechnique - Mathématiques
%D 2020
%P 93-142
%V 7
%I Ecole polytechnique
%U https://www.numdam.org/articles/10.5802/jep.112/
%R 10.5802/jep.112
%G en
%F JEP_2020__7__93_0

Chitour, Yacine; Mazanti, Guilherme; Sigalotti, Mario. Approximate and exact controllability of linear difference equations. Journal de l’École polytechnique - Mathématiques, Volume 7 (2020), pp. 93-142. doi : 10.5802/jep.112. https://www.numdam.org/articles/10.5802/jep.112/

References
Cited by

[1] de Avellar, Cerino E.; Hale, Jack K. On the zeros of exponential polynomials, J. Math. Anal. Appl., Volume 73 (1980) no. 2, pp. 434-452 | DOI | MR | Zbl

[2] Cassels, J. W. S. An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Physics, 45, Cambridge University Press, New York, 1957 | MR | Zbl

[3] Chitour, Yacine; Mazanti, Guilherme; Sigalotti, Mario Stability of non-autonomous difference equations with applications to transport and wave propagation on networks, Netw. Heterog. Media, Volume 11 (2016) no. 4, pp. 563-601 | DOI | MR | Zbl

[4] Chitour, Yacine; Mazanti, Guilherme; Sigalotti, Mario Persistently damped transport on a network of circles, Trans. Amer. Math. Soc., Volume 369 (2017) no. 6, pp. 3841-3881 | DOI | MR | Zbl

[5] Chyung, Dong Hak On the controllability of linear systems with delay in control, IEEE Trans. Automatic Control, Volume 15 (1970) no. 2, pp. 255-257 | DOI | MR

[6] Cooke, Kenneth L.; Krumme, David W. Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations, J. Math. Anal. Appl., Volume 24 (1968), pp. 372-387 | DOI | MR | Zbl

[7] Coron, Jean-Michel Control and nonlinearity, Math. Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007 | DOI | MR | Zbl

[8] Coron, Jean-Michel; Bastin, Georges; d’Andréa-Novel, Brigitte Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., Volume 47 (2008) no. 3, pp. 1460-1498 | DOI | MR | Zbl

[9] Coron, Jean-Michel; Nguyen, Hoai-Minh Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems, SIAM J. Math. Anal., Volume 47 (2015) no. 3, pp. 2220-2240 | DOI | MR | Zbl

[10] Cruz, Marianito A.; Hale, Jack K. Stability of functional differential equations of neutral type, J. Differential Equations, Volume 7 (1970), pp. 334-355 | DOI | MR | Zbl

[11] Datko, Richard Linear autonomous neutral differential equations in a Banach space, J. Differential Equations, Volume 25 (1977) no. 2, pp. 258-274 | DOI | MR | Zbl

[12] Diblík, Josef; Khusainov, Denys Ya.; Růžičková, M. Controllability of linear discrete systems with constant coefficients and pure delay, SIAM J. Control Optim., Volume 47 (2008) no. 3, pp. 1140-1149 | DOI | MR | Zbl

[13] Franklin, Gene F.; Powell, J. David; Workman, Michael L. Digital Control of Dynamic Systems, Addison-Wesley, 1997 | Zbl

[14] Fridman, Emilia; Mondié, Sabine; Saldivar, Belem Bounds on the response of a drilling pipe model, IMA J. Math. Control Inform., Volume 27 (2010) no. 4, pp. 513-526 | DOI | MR | Zbl

[15] Gohberg, I.; Shalom, T. On inversion of square matrices partitioned into nonsquare blocks, Integral Equations Operator Theory, Volume 12 (1989) no. 4, pp. 539-566 | DOI | MR | Zbl

[16] Hale, Jack K.; Infante, Ettore F.; Tsen, Fu Shiang Peter Stability in linear delay equations, J. Math. Anal. Appl., Volume 105 (1985) no. 2, pp. 533-555 | DOI | MR | Zbl

[17] Hale, Jack K.; Verduyn Lunel, Sjoerd M. Introduction to functional-differential equations, Applied Math. Sciences, 99, Springer-Verlag, New York, 1993, x+447 pages | DOI | MR | Zbl

[18] Hale, Jack K.; Verduyn Lunel, Sjoerd M. Strong stabilization of neutral functional differential equations, IMA J. Math. Control Inform., Volume 19 (2002) no. 1-2, pp. 5-23 | DOI | MR | Zbl

[19] Henry, Daniel Linear autonomous neutral functional differential equations, J. Differential Equations, Volume 15 (1974), pp. 106-128 | DOI | MR | Zbl

[20] Klöss, Bernd The flow approach for waves in networks, Oper. Matrices, Volume 6 (2012) no. 1, pp. 107-128 | DOI | MR | Zbl

[21] Lions, Jacques-Louis Contrôlabilité exacte des systèmes distribués, C. R. Acad. Sci. Paris Sér. I Math., Volume 302 (1986) no. 13, pp. 471-475 | DOI | Zbl

[22] Lions, Jacques-Louis Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., Volume 30 (1988) no. 1, pp. 1-68 | DOI | MR

[23] Mañé, Ricardo Ergodic theory and differentiable dynamics, Ergeb. Math. Grenzgeb. (3), 8, Springer-Verlag, Berlin, 1987 | DOI | MR | Zbl

[24] Mazanti, Guilherme Relative controllability of linear difference equations, SIAM J. Control Optim., Volume 55 (2017) no. 5, pp. 3132-3153 | DOI | MR | Zbl

[25] Melvin, William R. Stability properties of functional difference equations, J. Math. Anal. Appl., Volume 48 (1974), pp. 749-763 | DOI | MR | Zbl

[26] Michiels, Wim; Vyhlídal, Tomáš; Zítek, Pavel; Nijmeijer, Henk; Henrion, Didier Strong stability of neutral equations with an arbitrary delay dependency structure, SIAM J. Control Optim., Volume 48 (2009) no. 2, pp. 763-786 | DOI | MR | Zbl

[27] Ngoc, Pham Huu Anh; Huy, Nguyen Dinh Exponential stability of linear delay difference equations with continuous time, Vietnam J. Math., Volume 43 (2015) no. 2, pp. 195-205 | DOI | MR | Zbl

[28] O’Connor, Daniel A.; Tarn, T. J. On stabilization by state feedback for neutral differential-difference equations, IEEE Trans. Automatic Control, Volume 28 (1983) no. 5, pp. 615-618 | DOI | MR | Zbl

[29] O’Connor, Daniel A.; Tarn, T. J. On the function space controllability of linear neutral systems, SIAM J. Control Optim., Volume 21 (1983) no. 2, pp. 306-329 | DOI | MR | Zbl

[30] Pandolfi, L. Stabilization of neutral functional differential equations, J. Optimization Theory Appl., Volume 20 (1976) no. 2, pp. 191-204 | DOI | MR | Zbl

[31] Pospíšil, M; Diblík, J; Fečkan, M On relative controllability of delayed difference equations with multiple control functions, Proceedings of the International conference on numerical analysis and applied mathematics 2014 (ICNAAM-2014), Volume 1648, AIP Publishing, 2015, 130001 | DOI

[32] Rudin, Walter Real and complex analysis, McGraw-Hill Book Co., New York, 1987 | Zbl

[33] Rudin, Walter Functional analysis, International Series in Pure and Applied Math., McGraw-Hill, Inc., New York, 1991 | Zbl

[34] Salamon, Dietmar Control and observation of neutral systems, Research Notes in Math., 91, Pitman, Boston, MA, 1984 | MR | Zbl

[35] Slemrod, Marshall Nonexistence of oscillations in a nonlinear distributed network, J. Math. Anal. Appl., Volume 36 (1971), pp. 22-40 | DOI | MR | Zbl

[36] Sontag, Eduardo D. Mathematical control theory. Deterministic finite-dimensional systems, Texts in Applied Math., 6, Springer-Verlag, New York, 1998 | DOI | Zbl

[37] Viana, Marcelo Ergodic theory of interval exchange maps, Rev. Mat. Univ. Complut. Madrid, Volume 19 (2006) no. 1, pp. 7-100 | DOI | MR | Zbl

[38] Walters, Peter An introduction to ergodic theory, Graduate Texts in Math., 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl

Cited by Sources: