A Left Eigenvector Producing a Smooth Lyapunov Function of ISS Networks
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For a class of monotone nonlinear systems, it is shown that a continuously differentiable Lyapunov function can be constructed implicitly from a left eigenvector of vector fields. The left eigenvector which is a continuous function of state variables is deduced from a right eigenvector which represents a small gain condition. It is demonstrated that rounding off the edges of n-orthotopes, which is the maximization of state variables, yields level sets of the Lyapunov function. Applying the development to comparison systems gives continuously differentiable input-to-state Lyapunov functions of networks consisting of input-to-state systems which are not necessarily monotone.
KeywordsMonotone nonlinear systems Lyapunov functions Input-to-state stability Perron-Frobenius theory Small-gain condition
The work of H. Ito was supported in part by JSPS KAKENHI Grant Number 17K06499. B. S. Rüffer has been supported by ARC grant DP160102138.
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