Title: Null boundary controllability of a one-dimensional heat equation with internal point masses and variable coefficients Show Chinese title

Title: 一维热方程在内部点质量与变系数下的零边界能控性

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. Show Chinese abstract

Abstract: 在本文中，我们考虑一个由$N+1$个非齐次细杆组成的线性混合系统，这些细杆通过$N$个内部点质量连接，并且左端具有狄利克雷边界条件，右端具有狄利克雷控制。 利用详细的谱分析和矩理论，我们证明该系统在任何正时间$T$处都是零可控的。 为此，首先，我们采用朗斯基方法来获得与该系统相关的特征值$(\lambda_{n})_{n\in\N^*}$的特征方程。 其次，我们证明特征值$(\lambda_{n})_{n\in\N^*}$交错于具有齐次狄利克雷边界条件的$N+1$个解耦杆的特征值，并满足所谓的魏尔（Weyl）渐近公式。 最后，我们建立了特征值$(\lambda_{n})_{n\in\N^*}$的精确渐近估计。 作为结果，一方面，我们证明了谱间隙的统一下界。 在另一方面，我们推导了特征函数的$\mathcal{H}$-范数与其在右端的一阶导数之间的等价性。 作为我们谱分析的应用，我们还提出了关于带有内部点质量的薛定谔方程在左端具有狄利克雷控制的新可控性结果。