Metadata
Abstract
This article is devoted to the characterization of the minimal null control time for abstract linear control problem. More precisely we aim at giving a precise answer to the following question: what is the minimal time needed to drive the solution of the system starting from any initial condition in a given subspace to zero? Our setting will encompass a wide variety of systems of coupled one dimensional linear parabolic equations with a scalar control.
Following classical ideas we reduce this controllability issue to the resolution of a moment problem on the control and provide a new block resolution technique for this moment problem. The obtained estimates are sharp and hold uniformly for a certain class of operators. This uniformity allows various applications for parameter dependent control problems and permits us to deal naturally with the case of algebraically multiple eigenvalues in the underlying generator.
Our approach sheds light on a new phenomenon: the condensation of eigenvalues (which can cause a non zero minimal null control time in general) can be somehow compensated by the condensation of eigenvectors. We provide various examples (some are abstract systems, others are actual PDE systems) to highlight those new situations we are able to manage by the block resolution of the moment problem we propose.
References
[AB20] 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 | Article | MR 4097728
[ABM18] Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., Volume 140 (2018) no. 4, pp. 857-911 | Article | MR 3864704 | Zbl 1402.34086
[AI01] Riesz bases of exponentials and divided differences, Algebra Anal., Volume 13 (2001) no. 3, pp. 1-17 | MR 1850184
[AKBDK05] Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., Volume 11 (2005) no. 3, pp. 426-448 | Article | Numdam | MR 2148852 | Zbl 1125.93005
[AKBGBdT11] 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 | Article | MR 2851682
[AKBGBdT14] 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 | Article | MR 3250361 | Zbl 1304.35379
[AKBGBdT16] 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 | Article | MR 3535750 | Zbl 1342.93022
[AKBGBM19] Quantitative Fattorini–Hautus test and minimal null control time for parabolic problems, J. Math. Pures Appl., Volume 122 (2019), pp. 198-234 | Article | MR 3912641 | Zbl 1405.93032
[BB19] Boundary null-controllability of coupled parabolic systems with Robin conditions, Evol. Equ. Control Theory (2019), 42 pages | Article
[BC17] Heat equation on the Heisenberg group: observability and applications, J. Differ. Equations, Volume 262 (2017) no. 8, pp. 4475-4521 | Article | MR 3603278 | Zbl 1366.35206
[BCG14] Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 67-101 | Article | MR 3141729 | Zbl 1293.35148
[BDE20] Minimal time issues for the observability of Grushin-type equations, Ann. Inst. Fourier, Volume 70 (2020) no. 1, pp. 247-312 | Article | MR 4105940 | Zbl 07206863
[Ber33] Leçons sur les Progrès Récents de la Théorie des Séries de Dirichlet, Collection de monographie sur la théorie des fonctions, Gauthier–Villars, 1933 | Zbl 0008.11503
[BHHR15] Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., Volume 21 (2015) no. 2, pp. 487-512 | Article | MR 3348409 | Zbl 1311.93042
[BKL02] Ingham–Beurling type theorems with weakened gap conditions, Acta Math. Hung., Volume 97 (2002) no. 1-2, pp. 55-95 | Article | MR 1932795 | Zbl 1012.42022
[BM20] Analysis of non scalar control problems for parabolic systems by the block moment method (2020) (https://hal.archives-ouvertes.fr/hal-02397706, working paper)
[BMM15] 2D Grushin-type equations: minimal time and null controllable data, J. Differ. Equations, Volume 259 (2015) no. 11, pp. 5813-5845 | Article | MR 3397310 | Zbl 1321.35098
[BO14] 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 | Article | MR 3196788 | Zbl 1303.93036
[Cor07] Control and nonlinearity, Mathematical Surveys and Monographs, Volume 136, American Mathematical Society, 2007 | MR 2302744 | Zbl 1140.93002
[DK20] Control of the Grushin equation: non-rectangular control region and minimal time, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 3, 18 pages | Article | MR 4050579 | Zbl 07198266
[DM12] Resolvent conditions for the control of parabolic equations, J. Funct. Anal., Volume 263 (2012) no. 11, pp. 3641-3673 | Article | MR 2984079 | Zbl 1268.47050
[Dol73] Observability for the one-dimensional heat equation, Stud. Math., Volume 48 (1973), pp. 291-305 | Article | MR 333444 | Zbl 0264.35036
[Dup17] Controllability of a 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 | Article | MR 3716929 | Zbl 1375.93018
[Ego63] Some problems in the theory of optimal control, Zh. Vychisl. Mat. Mat. Fiz., Volume 3 (1963), pp. 887-904 | MR 157821
[Fat66] Some remarks on complete controllability, SIAM J. Control, Volume 4 (1966), pp. 686-694 | Article | MR 202484 | Zbl 0168.34906
[FCGBdT10] Boundary controllability of parabolic coupled equations, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1720-1758 | Article | MR 2665408 | Zbl 1196.93010
[FI96] Controllability of evolution equations, Lecture Notes Series, Seoul, Volume 34, Seoul National University, 1996 | MR 1406566
[FR71] Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292 | Article | MR 335014 | Zbl 0231.93003
[FR74] Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Q. Appl. Math., Volume 32 (1974), pp. 45-69 | Article | MR 510972 | Zbl 0281.35009
[Gal69] Optimal control of systems described by parabolic equations, SIAM J. Control, Volume 7 (1969), pp. 546-558 | Article | MR 264203 | Zbl 0188.47003
[Gau11] Problèmes d’Interpolation dans les Espaces de Paley–Wiener et Applications en Théorie du Contrôle (2011) (https://tel.archives-ouvertes.fr/tel-00652210) (Ph. D. Thesis)
[Jen94] Sur une expression simple du reste dans la formule d’interpolation de Newton, Kjöb. Overs. (1894), pp. 1-7 | Zbl 25.0401.01
[JPP07] Interpolation by vector-valued analytic functions, with applications to controllability, J. Funct. Anal., Volume 252 (2007) no. 2, pp. 517-549 | Article | MR 2360926 | Zbl 1137.46015
[JPP10] Weighted interpolation in Paley–Wiener spaces and finite-time controllability, J. Funct. Anal., Volume 259 (2010) no. 9, pp. 2424-2436 | Article | MR 2674120 | Zbl 1206.47020
[JPP13] Weighted multiple interpolation and the control of perturbed semigroup systems, J. Evol. Equ., Volume 13 (2013) no. 2, pp. 395-410 | Article | MR 3056309 | Zbl 1285.47021
[JPP14] Applications of Laplace–Carleson embeddings to admissibility and controllability, SIAM J. Control Optim., Volume 52 (2014) no. 2, pp. 1299-1313 | Article | MR 3196935 | Zbl 1294.30098
[Kir11] An introduction to the mathematical theory of inverse problems, Applied Mathematical Sciences, Volume 120, Springer, 2011 | Article | MR 3025302 | Zbl 1213.35004
[LR95] Contrôle exact de l’équation de la chaleur, Commun. Partial Differ. Equations, Volume 20 (1995) no. 1-2, pp. 335-356 | Article | Zbl 0819.35071
[LZ02] Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 19 (2002) no. 5, pp. 543-580 | Article | Numdam | MR 1922469
[Mil06] On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. Control Signals Syst., Volume 18 (2006) no. 3, pp. 260-271 | Article | MR 2272076 | Zbl 1105.93015
[Oli14] Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, Volume 3 (2014) no. 1, pp. 167-189 | Article | MR 3177264 | Zbl 1281.93024
[Oua20] Minimal time of null controllability of two parabolic equations, Math. Control Relat. Fields, Volume 10 (2020) no. 1, pp. 89-112 | Article | MR 4063628 | Zbl 07195056
[Pow81] Approximation theory and methods, Cambridge University Press, 1981 | Zbl 0453.41001
[PT87] Inverse spectral theory, Pure and Applied Mathematics, Volume 130, Academic Press Inc., 1987 | MR 894477 | Zbl 0623.34001
[Rud87] Real and complex analysis, McGraw-Hill, 1987 | Zbl 0925.00005
[Sam19] Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions. (2019) (https://arxiv.org/abs/1902.04472, preprint)
[Sch43] Étude des sommes d’exponentielles réelles, Actualités scientifiques et industrielles, Volume 959, Hermann, 1943 | MR 3533115 | Zbl 0061.13601
[Sha69] Overconvergence of Dirichlet series with complex exponents, J. Anal. Math., Volume 22 (1969), pp. 135-170 | Article | MR 274731 | Zbl 0177.10203
[TW09] Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, 2009 | Article | Zbl 1188.93002