A remark on the converging-input converging-state property

Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must also converge to the equilibrium. This "well-known" but hard-to-cite fact is proved and slightly generalized here.

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[2]  Eduardo Sontag Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[3]  Eduardo Sontag Remarks on stabilization and input-to-state stability , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.