Exact Boundary Controllability for the Boussinesq Equation with Variable Coefficients

TL;DR

This paper proves the exact boundary controllability of a variable-coefficient Boussinesq equation, using spectral analysis and the moment method, and extends results to the nonlinear case via fixed point techniques.

Contribution

It establishes the linear exact controllability for the variable-coefficient Boussinesq equation and demonstrates local controllability for the nonlinear version, advancing control theory for complex PDEs.

Findings

Linearized problem is exactly controllable in any positive time.

Spectral analysis and moment method are effective for controllability.

Local controllability is achieved for the nonlinear problem.

Abstract

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters: \begin{array}{lll} \rho(x)y_{tt}=-(\sigma(x)y_{xx})_{xx}+(q(x)y_x)_x-(y^2)_{xx},&&t>0,~x\in(0,l),\\ y(t,0)=\sigma(l)y_{xx}(t,0)=y(t,l)=0,~~\sigma(l)y_{xx}(t,l)=u(t)&&t>0, \end{array} where $l>0$, the coefficients $\rho(x)>0,\sigma(x)>0 $, $q(x)\geq0$ in $[0,l]$ and $u$ is the control acting at the end $x=l$. We prove that the linearized problem is exactly controllable in any time $T>0$. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument.