African Diaspora Journal of Mathematics

Interior Controllability of the Linear Beam Equation

H. Leiva and W. Pereira

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In this paper we prove the interior controllability of the Linear Beam Equation $$ \left\{ \begin{array}{ll} u_{tt}2\beta\Delta u_t + \Delta^2 u= 1_{\omega}u(t,x), & \mbox{in} \quad (0, \tau) \times \Omega,\\ u = \Delta u = 0, & \mbox{on} \quad (0, \tau) \times \partial \Omega, \end{array} \right. $$ where $\beta>1$, $\Omega$ is a sufficiently regular bounded domain in $\mathbb{R}^{N}$ $(N\geq 1)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$ and the distributed control $u\in L^{2}([0,\tau]; L^{2}(\Omega)).$ Specifically, we prove the following statement: For all $\tau >0$ the system is approximately controllable on $[0, \tau]$. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time $\tau >0$.

Article information

Afr. Diaspora J. Math. (N.S.), Volume 14, Number 1 (2012), 30-38.

First available in Project Euclid: 18 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93B05; 93C25

interior controllability Linear Beam Equation strongly continuous semigroups


Leiva , H.; Pereira , W. Interior Controllability of the Linear Beam Equation. Afr. Diaspora J. Math. (N.S.) 14 (2012), no. 1, 30--38.

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  • S. BADRAOUI,Approximate Controllability of a Reaction-Diffusion System with a Cross Diffusion Matrix and Fractional Derivatives on Bounded Domains. Journal of Boundary Value Problems, Vol. 2010(2010), Art. ID 281238, 14pgs.
  • R.F. CURTAIN, A.J. PRITCHARD, Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8. Springer Verlag, Berlin (1978).
  • H. LÁREZ, H. LEIVA AND J. UZCÁTEGUI, Controllability of Block Diagonal Systems and Applications. Int. J. Systems, Control and Communications, Vol. 3, No. 1, pp. 64-81.
  • H. LAREZ AND H. LEIVA, Interior controllability of a $2 \times 2$ reaction-diffusion system with cross-diffusion matrix. Boundary Value Problems, Vol. 2009, Article ID 560407, 9 pages, doi:10.1155/2009/560407.
  • H. LEIVA AND Y. QUINTANA, Interior controllability of a broad class of reaction diffusion equation. Mathematical Problems in Engineering, Vol. 2009, Article ID 708516, 8 pages, doi:10.1155/2009/708516.
  • H. LEIVA, A Lemma on $C_{0}$-Semigroups and Applications PDEs Systems. Quaestions Mathematicae, Vol. 26, pp. 247-265 (2003).
  • H. LEIVA AND N. MERENTES, Controllability of Second Order Equation in L2. Mathematical Problems in Engineering, Vol. 2010, Art. ID 147195, 11pages, doi:10.1155/2010/147195.
  • H. LEIVA AND N. MERENTES, Interior Controllability of the Thermoelastic Plate Equation. African Diaspora Journal of Mathematics, Vol. 12, No 1, pp. 1-14(2011).
  • H. LEIVA, N. MERENTES AND J.L. SANCHEZ, Interior Controllability of the nD Semilinear Heat Equation. African Diaspora Journal of Mathematics, Special Volume in Honor fo Profs. C. Corduneanu, A. Fink, and S. Zaidman, Vol. 12, Number 2, pp. 1-12 (2011).
  • M.H. PROTTER, Unique continuation for elliptic equations. Transaction of the American Mathematical Society, Vol. 95, No 1, Apr., 1960.
  • D. SEVICOVIC, Existence and limiting behaviour for damped nonlinear evolution equations with nonlocal terms. Comment. Math. Univ. Carolinae 31, 2, 283-293, 1990.