EXPONENTIAL STABILITY OF STRING SYSTEM WITH VARIABLE COEFFICIENTS UNDER NON-COLLOCATED FEEDBACK CONTROLS

In this paper, we describe a linear boundary non-collocated feedback controller to stabilize a string system with variable coefficients and investigate the exponential stability of the closed-loop system. We show that the system associates with a C-0 semigroup. Through the asymptotic analysis technique, we get the asymptotic values of all eigenvalues of the system operator A. Furthermore, we prove that there is a sequence of generalized eigenvectors of A that forms a Riesz basis with parentheses for the energy state space. Hence, the system satisfies the spectrum determined growth assumption. Finally, we show that the system is exponentially stable with suitable choice of the feedback gains.