Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method
Annales de l'Institut Fourier, Online first, 70 p.

In this article we study the null controllability of some abstract linear parabolic systems in tensor product spaces. This special structure allows us to reduce our controllability problem to a particular set of equations that looks like a moment problem, but that does not fall into the previous existing results of the literature.

We transform this non standard moment problem into an infinite family of more usual moment problems, yet coupled one from each other. This reformulation is done with enough care to ensure that the resulting set of equations can be solved, with suitable estimates, by using the recent “block moment method”. This is based on a careful analysis of the spectral structure of the underlying operator.

We notably apply our abstract result to show how strong the influence of geometry can be: we provide an example of boundary controlled parabolic system on a rectangle domain which is null controllable in arbitrary small time if two perpendicular faces of the boundary are controlled, whereas it is never null controllable if the control acts on only one face.

Dans cet article nous étudions la contrôlabilité à zéro d’une classe de systèmes paraboliques linéaires abstraits dans des produits tensoriels. Cette structure particulière nous permet de réduire la question de la contrôlabilité à un ensemble particulier d’équations qui ressemble à un problème de moments, mais qui ne relève pas des résultats existants de la littérature.

Nous transformons ce problème de moments non standard en une famille infinie de problèmes de moments usuels, mais couplés entre eux. Cette reformulation est choisie avec soin pour que le nouveau système obtenu puisse être résolu, avec des bonnes estimations des solutions, en utilisant la méthode des moments par blocs développée récemment. Tout ce travail est basé sur une analyse spectrale minutieuse de l’opérateur sous-jacent.

Nous appliquons notamment ce résultat abstrait pour montrer que la position du domaine de contrôle pour un problème de contrôle au bord de deux équations de la chaleur couplées peut être déterminante : nous donnons un exemple explicite d’un tel système posé sur un domaine rectangulaire en 2D qui est contrôlable à zéro en tout temps si le contrôle agit sur deux bords perpendiculaires du domaine mais qui n’est jamais contrôlable à zéro si le contrôle n’agit que sur un seul des bords du domaine.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3639
Classification: 93B05, 93C20, 93C25, 30E05, 35K90, 47A80
Keywords: Controllability, Parabolic systems, Geometric control conditions, Block moment method, Tensor products.
Mot clés : Contrôlabilité, Systèmes paraboliques, Conditions de contrôle géométrique, Méthode des moments par blocs, Produits tensoriels.
Boyer, Franck 1; Olive, Guillaume 2

1 Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9 (France)
2 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków (Poland)
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Boyer, Franck; Olive, Guillaume. Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method. Annales de l'Institut Fourier, Online first, 70 p.

[1] Alabau-Boussouira, Fatiha; Léautaud, Matthieu Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., Volume 99 (2013) no. 5, pp. 544-576 | DOI | MR | Zbl

[2] Allonsius, Damien; Boyer, Franck Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, Math. Control Relat. Fields, Volume 10 (2020) no. 2, pp. 217-256 | DOI | MR | Zbl

[3] Allonsius, Damien; Boyer, Franck; Morancey, Morgan Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., Volume 140 (2018) no. 4, pp. 857-911 | DOI | MR | Zbl

[4] Ammar-Khodja, Farid; Benabdallah, Assia; González-Burgos, Manuel; de Teresa, Luz The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., Volume 96 (2011) no. 6, pp. 555-590 | DOI | MR | Zbl

[5] Ammar-Khodja, Farid; Benabdallah, Assia; González-Burgos, Manuel; de Teresa, Luz Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal., Volume 267 (2014) no. 7, pp. 2077-2151 | DOI | MR | Zbl

[6] Ammar-Khodja, Farid; Benabdallah, Assia; González-Burgos, Manuel; de Teresa, Luz New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence, J. Math. Anal. Appl., Volume 444 (2016) no. 2, pp. 1071-1113 | DOI | MR | Zbl

[7] Badra, Mehdi; Takahashi, Takéo On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM, Control Optim. Calc. Var., Volume 20 (2014) no. 3, pp. 924-956 | DOI | Numdam | MR | Zbl

[8] Benabdallah, Assia; Boyer, Franck; González-Burgos, Manuel; Olive, Guillaume Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., Volume 52 (2014) no. 5, pp. 2970-3001 | DOI | MR | Zbl

[9] Benabdallah, Assia; Boyer, Franck; Morancey, Morgan A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. Henri Lebesgue, Volume 3 (2020), pp. 717-793 | DOI | Numdam | MR | Zbl

[10] Boyer, Franck Controllability of linear parabolic equations and systems, 2023 (lecture Notes, https://hal.archives-ouvertes.fr/hal-02470625v4)

[11] Boyer, Franck; Morancey, Morgan Analysis of non scalar control problems for parabolic systems by the block moment method, C. R. Math. Acad. Sci. Paris, Volume 361 (2023), pp. 1191-1248 | DOI | MR | Zbl

[12] Boyer, Franck; Olive, Guillaume Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields, Volume 4 (2014) no. 3, pp. 263-287 | DOI | MR | Zbl

[13] Carreño, Nicolás; Cerpa, Eduardo; Mercado, Alberto Boundary controllability of a cascade system coupling fourth- and second-order parabolic equations, Syst. Control Lett., Volume 133 (2019), 104542, 7 pages | DOI | MR | Zbl

[14] Coron, Jean-Michel Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007, xiv+426 pages | MR | Zbl

[15] Dolecki, Szymon Observability for the one-dimensional heat equation, Stud. Math., Volume 48 (1973), pp. 291-305 | DOI | MR | Zbl

[16] Duprez, Michel Controllability of a 2×2 parabolic system by one force with space-dependent coupling term of order one, ESAIM, Control Optim. Calc. Var., Volume 23 (2017) no. 4, pp. 1473-1498 | DOI | Numdam | MR | Zbl

[17] Egorov, Ju. V. Some problems in the theory of optimal control, Zh. Vychisl. Mat. Mat. Fiz., Volume 3 (1963), pp. 887-904 | MR | Zbl

[18] Engel, Klaus-Jochen; Nagel, Rainer One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer, 2000, xxii+586 pages (with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt) | MR | Zbl

[19] Fattorini, Hector O. Some remarks on complete controllability, SIAM J. Control, Volume 4 (1966), pp. 686-694 | DOI | MR | Zbl

[20] Fattorini, Hector O. Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, Volume 13 (1975), pp. 1-13 | DOI | MR | Zbl

[21] Fattorini, Hector O.; Russell, David L. Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292 | DOI | MR | Zbl

[22] Fernández-Cara, Enrique; González-Burgos, Manuel; de Teresa, Luz Boundary controllability of parabolic coupled equations, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1720-1758 | DOI | MR | Zbl

[23] Gal’chuk, Leonid I. Optimal control of systems described by parabolic equations, SIAM J. Control, Volume 7 (1969), pp. 546-558 | DOI | MR | Zbl

[24] Kato, Tosio Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 1966, xix+592 pages | MR | Zbl

[25] Lebeau, Gilles; Robbiano, Luc Contrôle exact de l’équation de la chaleur, Commun. Partial Differ. Equations, Volume 20 (1995) no. 1-2, pp. 335-356 | DOI | MR | Zbl

[26] Lebeau, Gilles; Zuazua, Enrique Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 4, pp. 297-329 | DOI | MR | Zbl

[27] Luca, Florian; de Teresa, Luz Control of coupled parabolic systems and Diophantine approximations, SeMA J., Volume 61 (2013), pp. 1-17 | DOI | MR | Zbl

[28] Miller, Luc On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math., Volume 129 (2005) no. 2, pp. 175-185 | DOI | MR | Zbl

[29] Miller, Luc A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst., Ser. B, Volume 14 (2010) no. 4, pp. 1465-1485 | DOI | MR | Zbl

[30] Olive, Guillaume Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, Volume 3 (2014) no. 1, pp. 167-189 | DOI | MR | Zbl

[31] Ouaili, Lydia Minimal time of null controllability of two parabolic equations, Math. Control Relat. Fields, Volume 10 (2020) no. 1, pp. 89-112 http://aimsciences.org//article/id/91183c77-31ff-402d-a7cd-fec1c660c31d | DOI | MR | Zbl

[32] Reed, Michael; Simon, Barry Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, 1978, xv+396 pages | MR | Zbl

[33] Reed, Michael; Simon, Barry Methods of modern mathematical physics. I. Functional analysis, Academic Press Inc., 1980, xv+400 pages | MR | Zbl

[34] Samb, El Hadji Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions, ESAIM, Control Optim. Calc. Var., Volume 27 (2021) no. suppl., S29, 43 pages | DOI | MR | Zbl

[35] Tenenbaum, Gérald; Tucsnak, Marius On the null-controllability of diffusion equations, ESAIM, Control Optim. Calc. Var., Volume 17 (2011) no. 4, pp. 1088-1100 | DOI | Numdam | MR | Zbl

[36] West, Douglas B. Introduction to graph theory, Prentice Hall, Inc., 2001, xx+588 pages

[37] Young, Robert M. An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, 93, Academic Press Inc., 1980, x+246 pages | MR | Zbl

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