Time optimal control problems for some non-smooth systems
MCRF Home
September 2014, 4(3): 263-287. doi: 10.3934/mcrf.2014.4.263
Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients
Franck Boyer 1, and Guillaume Olive 1,
Aix-Marseille Université, CNRS, Centrale Marseille, Laboratoire d'Analyse Topologie et Probabilités, UMR 7353, 13453 Marseille, France, France
Received August 2013 Revised December 2013 Published April 2014
In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms.
Our analysis is mainly based on a criterion given by Fattorini in [12] (and systematically used in [22] for instance), that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems. This work gives theoretical justifications of some numerical observations described in [9].
Keywords: Hautus test., distributed controllability, Parabolic systems, unique continuation, geometric condition.
Mathematics Subject Classification: 93B05, 93B07, 93C05, 35K0.
Citation: Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263
F. Alabau-Boussouira, Controllability of cascade coupled systems of multi-dimensional evolution PDEs by a reduced number of controls,, C. R. Math. Acad. Sci. Paris, 350 (2012), 577. doi: 10.1016/j.crma.2012.05.009. Google Scholar
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications,, J. Math. Pures Appl., 99 (2013), 544. doi: 10.1016/j.matpur.2012.09.012. Google Scholar
G. Alessandrini and L. Escauriaza, Null-controllability of one-dimensional parabolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 284. doi: 10.1051/cocv:2007055. Google Scholar
F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems,, Differ. Equ. Appl., 1 (2009), 427. doi: 10.7153/dea-01-24. Google Scholar
F. Ammar-Khodja, A. Benabdallah, C. Dupai and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems,, J. Evol. Equ., 9 (2009), 267. doi: 10.1007/s00028-009-0008-8. Google Scholar
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267. doi: 10.3934/mcrf.2011.1.267. Google Scholar
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences,, preprint, (2013). Google Scholar
A. Benabdallah, M. Cristofol, P. Gaitan and L. de Teresa, Controllability to trajectories for some parabolic systems of three and two equations by one control force,, Math. Control Relat. Fields, 4 (2014), 17. doi: 10.3934/mcrf.2014.4.17. Google Scholar
F. Boyer, On the penalized hum approach and its applications to the numerical approximation of null-controls for parabolic problems,, ESAIM Proceedings, 41 (2013), 15. doi: 10.1051/proc/201341002. Google Scholar
M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems,, preprint, (2012). Google Scholar
J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (2007). Google Scholar
H. O. Fattorini, Some remarks on complete controllability,, SIAM J. Control, 4 (1966), 686. doi: 10.1137/0304048. Google Scholar
E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations,, J. Funct. Anal., 259 (2010), 1720. doi: 10.1016/j.jfa.2010.06.003. Google Scholar
A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations,, Lecture Notes, (1996). Google Scholar
M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic {PDE}s by one control force,, Port. Math., 67 (2010), 91. doi: 10.4171/PM/1859. Google Scholar
J.-M. Ghidaglia, Some backward uniqueness results,, Nonlinear Anal., 10 (1986), 777. doi: 10.1016/0362-546X(86)90037-4. Google Scholar
O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations,, ESAIM Control Optim. Calc. Var., 16 (2010), 247. doi: 10.1051/cocv/2008077. Google Scholar
F. Luca and L. de Teresa, Control of coupled parabolic systems and Diophantine approximations,, S$\vec e$MA J., 61 (2013), 1. doi: 10.1007/s40324-013-0004-3. Google Scholar
A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils,, Translated from the Russian by H. H. McFaden, (1988). Google Scholar
K. Mauffrey, On the null controllability of a $3\times3$ parabolic system with non-constant coefficients by one or two control forces,, J. Math. Pures Appl., 99 (2013), 187. doi: 10.1016/j.matpur.2012.06.010. Google Scholar
S. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques,, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 31 (1958), 219. Google Scholar
G. Olive, Boundary approximate controllability of some linear parabolic systems,, Evol. Equ. Control Theory, 3 (2014), 167. doi: 10.3934/eect.2014.3.167. Google Scholar
L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291. doi: 10.1016/j.crma.2011.01.014. Google Scholar
Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control & Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165
José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
Brooke L. Hollingsworth, R.E. Showalter. Semilinear degenerate parabolic systems and distributed capacitance models. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 59-76. doi: 10.3934/dcds.1995.1.59
Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control & Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27
Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems & Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309
Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047
Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control & Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465
Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167
Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control & Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems & Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619
Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827
Roberto Triggiani. Unique continuation of boundary over-determined Stokes and Oseen eigenproblems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 645-677. doi: 10.3934/dcdss.2009.2.645
Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613
Franck Boyer Guillaume Olive