JournalspmVol. 68, No. 3pp. 279–296

On the stabilization and controllability for a third order linear equation

  • Patrícia Nunes da Silva

    Universidade do Estado do Rio de Janeiro, Rio De Janeiro, Brazil
  • Carlos Frederico Vasconcellos

    Universidade do Estado do Rio de Janeiro, Rio De Janeiro, Brazil
On the stabilization and controllability for a third order linear  equation cover

A subscription is required to access this article.

Abstract

We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation:

iut+iγux+αuxx+iβuxxx=0,iu_t+i\gamma u_x+ \alpha u_{xx} + i\beta u_{xxx} =0,

where u=u(x,t)u=u(x,t) is a complex valued function defined in (0,L)×(0,+)(0,L)\times(0,+\infty) and α\alpha, β\beta and γ\gamma are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths LL, named X\mathcal{X}, in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters α\alpha, β\beta and γ\gamma.

Cite this article

Patrícia Nunes da Silva, Carlos Frederico Vasconcellos, On the stabilization and controllability for a third order linear equation. Port. Math. 68 (2011), no. 3, pp. 279–296

DOI 10.4171/PM/1892