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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 aDirichlet boundary condition on the left end, and Dirichlet control on theright end. Using a detailed spectral analysis and the moment theory, we provethat this system is null controllable at any positive time $T$. To this end,firstly, we implement the Wronskian technique to obtain the characteristicequation for the eigenvalues $(\lambda_{n})_{n\in\N^*}$ associated with thissystem. Secondly, we provide that the eigenvalues $(\lambda_{n})_{n\in\N^*}$interlace those of the $N+1$ decoupled rods with homogeneous Dirichlet boundaryconditions, and satisfy the so-called Weyl's asymptotic formula. Finally, weestablish sharp asymptotic estimates of the eigenvalues$(\lambda_{n})_{n\in\N^*}$. As consequence, on one hand, we prove a uniformlower bound for the spectral gap. On another hand, we derive the equivalencebetween the $\mathcal{H}$-norm of the eigenfunctions and their first derivativeat the right end. As an application of our spectral analysis, we also presentnew controllability result for the Schr\"{o}dinger equation with an internalpoint mass and Dirichlet control on the left end.

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