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

Abstract

In this paper, we consider a linear hybrid system which is composed of $N+1$ non-homogeneous thin rods connected by $N$ interior-point masses with a Dirichlet boundary condition on the left end, and Dirichlet control on the right end. Using a detailed spectral analysis and the moment theory, we prove that this system is null controllable at any positive time $T$. To this end, firstly, we implement the Wronskian technique to obtain the characteristic equation for the eigenvalues $({\lambda }_n{)}_{n\in {\mathbb{N}}^{\ast }}$ associated with this system. Secondly, we provide that the eigenvalues $({\lambda }_n{)}_{n\in {\mathbb{N}}^{\ast }}$ interlace those of the $N+1$ decoupled rods with homogeneous Dirichlet boundary conditions, and satisfy the so-called Weyl's asymptotic formula. Finally, we establish sharp asymptotic estimates of the eigenvalues $({\lambda }_n{)}_{n\in {\mathbb{N}}^{\ast }}$. As consequence, on one hand, we prove a uniform lower bound for the spectral gap. On another hand, we derive the equivalence between the $\mathcal{H}$-norm of the eigenfunctions and their first derivative at the right end. As an application of our spectral analysis, we also present new controllability result for the Schrödinger equation with an internal point mass and Dirichlet control on the left end.