Null boundary controllability of a one-dimensional heat equation with an internal point mass and variable coefficients

TL;DR

This paper proves the null boundary controllability of a one-dimensional heat equation system with an internal point mass and variable coefficients, using spectral analysis and the moment method, without restrictive coefficient conditions.

Contribution

It establishes null controllability for a hybrid heat system with variable coefficients and an internal mass, expanding control theory for complex PDE systems.

Findings

System is null controllable with boundary controls

Spectral gap is positive without extra coefficient restrictions

Method applies to both Dirichlet and Neumann controls

Abstract

In this paper we consider a linear hybrid system which composed by two non-homogeneous rods connected by a point mass and generated by the equation\bea\left\{ \begin{array}{ll} \rho_{1}(x)u_{t}=(\sigma_{1}(x)u_{x})_{x}-q_{1}(x)u,& x\in(-1,0),~t>0, \rho_{2}(x)v_{t}=(\sigma_{2}(x)v_{x})_{x}-q_{2}(x)v,& x\in(0,1), ~~~t>0, M z_{t}(t)=\sigma_{2}(0)v_{x}(0,t)-\sigma_{1}(0)u_{x}(0,t), &t>0, \end{array} \right. \eea with Dirichlet boundary condition on the left end $x=-1$ and a boundary control acts on the right end $x=1$. We prove that this system is null controllable with Dirichlet or Neumann boundary controls. Our approach is mainly based on a detailed spectral analysis together with the moment method. In particular, we show that the associated spectral gap in both cases (Dirichlet or Neumann boundary controls) are positive without further conditions on the coefficients $\rho_{i}$, $\sigma_{i}$ and $q_{i}$ $(i=1,2)$ other than the regularities.