In this paper, we study approximate and exact controllability of the linear difference equation in , with and , using as a basic tool a representation formula for its solution in terms of the initial condition, the control , and some suitable matrix coefficients. When 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 . 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 dans , avec et , en s’appuyant sur une formule de représentation de la solution en termes de la condition initiale, du contrôle et de coefficients matriciels appropriés. Lorsque 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 . Le résultat correspondant à la contrôlabilité exacte est vrai au moins pour les systèmes bi-dimensionnels avec deux retards.
Accepted:
Published online:
DOI: 10.5802/jep.112
Keywords: Linear difference equation, delay, approximate controllability, exact controllability
Mot clés : Équation aux différences linéaire, retard, contrôlabilité approchée, contrôlabilité exacte
Yacine Chitour 1; Guilherme Mazanti 2; Mario Sigalotti 3
@article{JEP_2020__7__93_0, author = {Yacine Chitour and Guilherme Mazanti and Mario Sigalotti}, title = {Approximate and exact controllability of linear difference equations}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} 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://jep.centre-mersenne.org/articles/10.5802/jep.112/} }
TY - JOUR AU - Yacine Chitour AU - Guilherme Mazanti AU - Mario Sigalotti 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.112/ DO - 10.5802/jep.112 LA - en ID - JEP_2020__7__93_0 ER -
%0 Journal Article %A Yacine Chitour %A Guilherme Mazanti %A Mario Sigalotti %T Approximate and exact controllability of linear difference equations %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 93-142 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.112/ %R 10.5802/jep.112 %G en %F JEP_2020__7__93_0
Yacine Chitour; Guilherme Mazanti; Mario Sigalotti. 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://jep.centre-mersenne.org/articles/10.5802/jep.112/
[1] - “On the zeros of exponential polynomials”, J. Math. Anal. Appl. 73 (1980) no. 2, p. 434-452 | DOI | MR | Zbl
[2] - An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Physics, vol. 45, Cambridge University Press, New York, 1957 | MR | Zbl
[3] - “Stability of non-autonomous difference equations with applications to transport and wave propagation on networks”, Netw. Heterog. Media 11 (2016) no. 4, p. 563-601 | DOI | MR | Zbl
[4] - “Persistently damped transport on a network of circles”, Trans. Amer. Math. Soc. 369 (2017) no. 6, p. 3841-3881 | DOI | MR | Zbl
[5] - “On the controllability of linear systems with delay in control”, IEEE Trans. Automatic Control 15 (1970) no. 2, p. 255-257 | DOI | MR
[6] - “Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations”, J. Math. Anal. Appl. 24 (1968), p. 372-387 | DOI | MR | Zbl
[7] - Control and nonlinearity, Math. Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | DOI | MR | Zbl
[8] - “Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems”, SIAM J. Control Optim. 47 (2008) no. 3, p. 1460-1498 | DOI | MR | Zbl
[9] - “Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems”, SIAM J. Math. Anal. 47 (2015) no. 3, p. 2220-2240 | DOI | MR | Zbl
[10] - “Stability of functional differential equations of neutral type”, J. Differential Equations 7 (1970), p. 334-355 | DOI | MR | Zbl
[11] - “Linear autonomous neutral differential equations in a Banach space”, J. Differential Equations 25 (1977) no. 2, p. 258-274 | DOI | MR | Zbl
[12] - “Controllability of linear discrete systems with constant coefficients and pure delay”, SIAM J. Control Optim. 47 (2008) no. 3, p. 1140-1149 | DOI | MR | Zbl
[13] - Digital Control of Dynamic Systems, Addison-Wesley, 1997 | Zbl
[14] - “Bounds on the response of a drilling pipe model”, IMA J. Math. Control Inform. 27 (2010) no. 4, p. 513-526 | DOI | MR | Zbl
[15] - “On inversion of square matrices partitioned into nonsquare blocks”, Integral Equations Operator Theory 12 (1989) no. 4, p. 539-566 | DOI | MR | Zbl
[16] - “Stability in linear delay equations”, J. Math. Anal. Appl. 105 (1985) no. 2, p. 533-555 | DOI | MR | Zbl
[17] - Introduction to functional-differential equations, Applied Math. Sciences, vol. 99, Springer-Verlag, New York, 1993 | DOI | MR | Zbl
[18] - “Strong stabilization of neutral functional differential equations”, IMA J. Math. Control Inform. 19 (2002) no. 1-2, p. 5-23 | DOI | MR | Zbl
[19] - “Linear autonomous neutral functional differential equations”, J. Differential Equations 15 (1974), p. 106-128 | DOI | MR | Zbl
[20] - “The flow approach for waves in networks”, Oper. Matrices 6 (2012) no. 1, p. 107-128 | DOI | MR | Zbl
[21] - “Contrôlabilité exacte des systèmes distribués”, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986) no. 13, p. 471-475 | DOI | Zbl
[22] - “Exact controllability, stabilization and perturbations for distributed systems”, SIAM Rev. 30 (1988) no. 1, p. 1-68 | DOI | MR
[23] - Ergodic theory and differentiable dynamics, Ergeb. Math. Grenzgeb. (3), vol. 8, Springer-Verlag, Berlin, 1987 | DOI | MR | Zbl
[24] - “Relative controllability of linear difference equations”, SIAM J. Control Optim. 55 (2017) no. 5, p. 3132-3153 | DOI | MR | Zbl
[25] - “Stability properties of functional difference equations”, J. Math. Anal. Appl. 48 (1974), p. 749-763 | DOI | MR | Zbl
[26] - “Strong stability of neutral equations with an arbitrary delay dependency structure”, SIAM J. Control Optim. 48 (2009) no. 2, p. 763-786 | DOI | MR | Zbl
[27] - “Exponential stability of linear delay difference equations with continuous time”, Vietnam J. Math. 43 (2015) no. 2, p. 195-205 | DOI | MR | Zbl
[28] - “On stabilization by state feedback for neutral differential-difference equations”, IEEE Trans. Automatic Control 28 (1983) no. 5, p. 615-618 | DOI | MR | Zbl
[29] - “On the function space controllability of linear neutral systems”, SIAM J. Control Optim. 21 (1983) no. 2, p. 306-329 | DOI | MR | Zbl
[30] - “Stabilization of neutral functional differential equations”, J. Optimization Theory Appl. 20 (1976) no. 2, p. 191-204 | DOI | MR | Zbl
[31] - “On relative controllability of delayed difference equations with multiple control functions”, in Proceedings of the International conference on numerical analysis and applied mathematics 2014 (ICNAAM-2014), vol. 1648, AIP Publishing, 2015, article ID 130001 | DOI
[32] - Real and complex analysis, McGraw-Hill Book Co., New York, 1987 | Zbl
[33] - Functional analysis, International Series in Pure and Applied Math., McGraw-Hill, Inc., New York, 1991 | Zbl
[34] - Control and observation of neutral systems, Research Notes in Math., vol. 91, Pitman, Boston, MA, 1984 | MR | Zbl
[35] - “Nonexistence of oscillations in a nonlinear distributed network”, J. Math. Anal. Appl. 36 (1971), p. 22-40 | DOI | MR | Zbl
[36] - Mathematical control theory. Deterministic finite-dimensional systems, Texts in Applied Math., vol. 6, Springer-Verlag, New York, 1998 | DOI | Zbl
[37] - “Ergodic theory of interval exchange maps”, Rev. Mat. Univ. Complut. Madrid 19 (2006) no. 1, p. 7-100 | DOI | MR | Zbl
[38] - An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl
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