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On the decay rate for degenerate gradient flows subject to persistent excitation

Abstract : In this paper, we study the worst rate of exponential decay for degenerate gradient flows in R n of the formẋ(t) = −c(t)c(t) x(t), issued from adaptative control theory, under a persistent excitation (PE) condition. That is, there exists a, b, T > 0 such that, for every t ≥ 0 it holds a Id n ≤ ∫ t+T t c(s)c(s) ds ≤ b Id n. Our main result is an upper bound of the form a (1+b) 2 T , to be compared with the well-known lower bounds of the form a (1+nb 2)T. As a byproduct, we also provide necessary conditions for the exponential convergence of these systems under a more general (PE) condition. Our techniques relate the worst rate of exponential decay to an optimal control problem that we study in detail.
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https://hal.archives-ouvertes.fr/hal-03020089
Contributor : Dario Prandi <>
Submitted on : Monday, November 23, 2020 - 4:44:12 PM
Last modification on : Wednesday, December 2, 2020 - 3:40:18 AM

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  • HAL Id : hal-03020089, version 1

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Yacine Chitour, Paolo Mason, Dario Prandi. On the decay rate for degenerate gradient flows subject to persistent excitation. IFAC World Congress 2020, Germany, Jul 2020, Berlin, Germany. ⟨hal-03020089⟩

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