Boundary Controllability of Two Vibrating Strings Connected by a Point Mass with Variable Coefficients

Abstract

S. Hansen and E. Zuazua [SIAM J. Control Optim., 33 (1995), pp. 1357--1391] studied the problem of exact controllability of two strings connected by a point mass with constant physical coefficients. In this paper we study the same problem with variable physical coefficients. This system is generated by the following equations: $\rho(x) u_{tt}=(\sigma(x) u_{x})_{x}-q(x)u$, $x\in (-1,0)\cup (0,1)$, $t>0$, $Mu_{tt}(0,t)+\sigma_{1}(0)u_{x}(0^{-},t)-\sigma_{2}(0)u_{x}(0^{+},t)=0$, $t>0$, with a Dirichlet boundary condition on the left end and a control on the right end. We prove that this system is exactly controllable in an asymmetric space for the control time $T> 2\int_{-1}^{1}(\frac{\rho(x)}{\sigma(x)})^{\frac{1}{2}}dx$. We establish the equivalence between a suitable asymmetric norm of the initial data and the $L^{2}(0,T)$-norm of $u_{x}(1,t)$ (where $u$ is the solution of the uncontrolled system). Our approach is mainly based on a detailed spectral analysis and the theory of divided differences. In particular, we prove that the spectral gap $(\sqrt{\lambda_{n+1}}-\sqrt{\lambda_{n}})$ tends to zero of the order of $\frac{1}{n}$.