Vector Extensions of Halanay’s Inequality

We provide two extensions of Halanay’s inequality, where the scalar function in the usual Halanay’s inequality is replaced by a vector valued function, under a Metzler condition. We provide an easily checked necessary and sufficient condition for asymptotic convergence of the function to the zero vector in the time-invariant case. For the time-varying cases, we provide a sufficient condition for this convergence, which can be easily checked when the systems are periodic. We illustrate our results in cases that are beyond the scope of prior asymptotic stability results.

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