Exponential and Super Stability of a Wave Network
In this paper, we investigate the spectral distribution and stability of a star-shaped wave network with N edges, of which the feedback gain constants fail to satisfy the assumptions for Riesz basis generation. By a detailed spectral analysis, we present the explicit expressions of the spectra, which consist of simple eigenvalues located on a vertical line in the complex left half-plane. In addition we show that the eigenvectors are not complete in the state space. Further, we decompose the state space into the spectral-subspace and another invariant subspace of infinite dimension, which form a topological direct sum. We prove that, in the spectral-subspace, the solution can be expanded according to the eigenvectors, and hence the solution is exponentially stable; in the other subspace, the associated semigroup is super-stable, i.e., the solution is identical to zero after finite time. In particular, we give the explicit decay rate and the maximum existence time of the nonzero part of the solution.
wave network completeness exponentially stable spectrum super-stable