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

TL;DR

This paper proves null controllability for a complex one-dimensional heat system with internal point masses and variable coefficients using spectral analysis and moment theory, and extends results to a Schrödinger equation.

Contribution

It introduces a detailed spectral analysis and moment theory approach to establish null controllability for a hybrid heat system with internal masses and variable coefficients, including eigenvalue asymptotics.

Findings

System is null controllable at any positive time T.

Eigenvalues interlace with those of decoupled rods and satisfy Weyl's asymptotic formula.

Established spectral gap bounds and eigenfunction estimates.

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 $(\lambda_{n})_{n\in\N^*}$ associated with this system. Secondly, we provide that the eigenvalues $(\lambda_{n})_{n\in\N^*}$ 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 $(\lambda_{n})_{n\in\N^*}$. 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 Schr\"{o}dinger equation with an internal point mass and Dirichlet control on the left end.