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.

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