BOUNDARY CONTROLLABILITY OF TWO VIBRATING STRINGS CONNECTED BY A POINT MASS WITH VARIABLE COEFFICIENTS

摘要

Hansen and E. Zuazua [SIAM T. 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 is an element of (-1, 0)boolean OR(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 integral(1)(-1) (rho(x)/sigma(x))(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 (root lambda(n+1)-root lambda(n)) tends to zero of the order of 1/n.