2004, 11(2&3): 311-324. doi: 10.3934/dcds.2004.11.311

Controllability properties of a vibrating string with variable axial load

1. 

Department of Mathematics, Washington State University, Pullman, WA 99164-3113, United States

Received  October 2002 Revised  February 2004 Published  June 2004

We show that the set of equilibrium-like states $ (y_d, 0) $ of a vibrating string which can approximately be reached in the energy space $ H_0^1 (0,1) \times L^2 (0,1) $ from almost any non-zero initial datum by varying its axial load is dense in the subspace $ H_0^1 (0,1) \times $ {0} of this space. Our result is based on a constructive argument and makes use of piecewise constant-in-time control functions (loads) only, which enter the model equation as coefficients.
Citation: Alexander Khapalov. Controllability properties of a vibrating string with variable axial load. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 311-324. doi: 10.3934/dcds.2004.11.311
[1]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[2]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125

[3]

Sergei A. Avdonin, Boris P. Belinskiy. On controllability of a linear elastic beam with memory under longitudinal load. Evolution Equations & Control Theory, 2014, 3 (2) : 231-245. doi: 10.3934/eect.2014.3.231

[4]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

[5]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[6]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[7]

Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control & Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161

[8]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[9]

Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034

[10]

Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311

[11]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[12]

Heinz Schättler, Urszula Ledzewicz. Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4611-4638. doi: 10.3934/dcds.2015.35.4611

[13]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[14]

Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721

[15]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014

[16]

Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247

[17]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028

[18]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429

[19]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

[20]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2018, 8 (0) : 1-2. doi: 10.3934/mcrf.2019006

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]