  S. Hansen and E. Zuazua [SIAM J. Cont. Optim., 1995] 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 Dirichlet boundary condition on the left end and a control acts 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 $u_{x}(1,t)$. Our approach is mainly based on a detailed
spectral analysis and the theory of divided differences. More precisely, we
prove that the spectral gap tends to zero with a precise asymptotic estimate.
