The Spectral Analysis and Exponential Stability of a 1-d 2 × 2 Hyperbolic System with Proportional Feedback Control

In this paper, the well-posedness and stability of a 1-d 2 × 2 hyperbolic system under proportional feedback control is addressed. The semigroup approach and norm equivalence theorem is adopted to obtain the explicit dissipative condition for control parameters. The asymptotic spectral analysis of the system is presented. It is shown that the real part of the eigenvalues goes to some negative constant as the modulus of eigenvalues goes to +∞. Moreover, we find that there is a group of generalized eigenfunctions which forms a Riesz basis of the Hilbert state space. Hence, the spectrum determined growth condition holds, and then the exponential stability is established.