October  2006, 14(4): 643-672. doi: 10.3934/dcds.2006.14.643

On conditions that prevent steady-state controllability of certain linear partial differential equations

1. 

Laboratoire des systèmes et signaux, Université Paris-Sud, CNRS, Supélec, 91192, Gif-sur-Yvette, France

2. 

Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405, Orsay, France

3. 

Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53 - Edificio U5, 20125 - Milano, Italy

Received  October 2004 Revised  August 2005 Published  January 2006

In this paper, we investigate the connections between controllability properties of distributed systems and existence of non zero entire functions subject to restrictions on their growth and on their sets of zeros. Exploiting these connections, we first show that, for generic bounded open domains in dimension $n\geq 2$, the steady--state controllability for the heat equation with boundary controls dependent only on time, does not hold. In a second step, we study a model of a water tank whose dynamics is given by a wave equation on a two-dimensional bounded open domain. We provide a condition which prevents steady-state controllability of such a system, where the control acts on the boundary and is only dependent on time. Using that condition, we prove that the steady-state controllability does not hold for generic tank shapes.
Citation: Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643
[1]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[2]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039

[3]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations and Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023

[4]

Umberto Biccari, Mahamadi Warma, Enrique Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1949-1978. doi: 10.3934/cpaa.2020086

[5]

Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177

[6]

Piermarco Cannarsa, Alessandro Duca, Cristina Urbani. Exact controllability to eigensolutions of the bilinear heat equation on compact networks. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1377-1401. doi: 10.3934/dcdss.2022011

[7]

Abdelaziz Khoutaibi, Lahcen Maniar, Omar Oukdach. Null controllability for semilinear heat equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1525-1546. doi: 10.3934/dcdss.2022087

[8]

Salah-Eddine Chorfi, Ghita El Guermai, Lahcen Maniar, Walid Zouhair. Impulse null approximate controllability for heat equation with dynamic boundary conditions. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022026

[9]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[10]

Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042

[11]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1

[12]

William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020

[13]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

[14]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[15]

Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations and Control Theory, 2022, 11 (4) : 1285-1307. doi: 10.3934/eect.2021044

[16]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078

[17]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure and Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

[18]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[19]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[20]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (5)

[Back to Top]