Boundary Controllability Of Two Vibrating Strings Connected By A Point Mass With Variable Coefficients

Abstract

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 $$\rho (x)u_{tt}=(\sigma (x)u_x{)}_x-q(x)u,x\in (-1,0)\cup (0,1),t>0,$$ $$M{u}_{tt}(0,t)+{\sigma }_1(0)u_x(0^{-},t)-{\sigma }_2(0)u_x(0^{+},t)=0,t>0,$$ 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.

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