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If the Lyapunov function has a constant limit for t →∞, should its derivative always strive to zero?
Itzhak Barkana

Last modified: 2023-06-29


This work attempts to explain the importance of providing the appropriate answer to this, apparently purely theoretical question, for the important topic of nonlinear systems stability analysis. The Lyapunov stability analysis is the methodology most commonly used to guarantee of stability of adaptive control systems and of nonlinear systems in general. The main idea of this methodology is the relationship between the ultimate behavior of a function, called Lyapunov function, and the ultimate behavior of its derivative. The proofs of stability attempt to show that the trajectories of a system end such that the Lyapunov function ultimately reaches a constant value, while the Lyapunov derivative ultimately tends to zero. However, counterexamples have been suggested, which apparently cancel the validity of those proofs of stability. The importance of this issue should explain the title of this paper, as the routine answer to the question in the title seems to be negative. In other words, the claim is that a function can reach a finite limit, while its derivative can keep moving up-and-down forever. Because the implications of a negative answer could simply leave the impression that the derivative of a ultimately constant function can actually be anything and not necessarily zero, and because of its negative implications to stability analysis, we decided to thoroughly review the computations that lead to the above routine answer. This work intends to show that correcting a few common misuses of some Calculus rules can provide the proper answer to this intriguing question. After reviewing a few complex counterexamples, it is shown in this work that, going back and reviewing the correct use of the basic rules for limit and derivative does provide the only correct answer: yes, if a function ends with a finite (not necessarily zero) constant limit, its derivative ends with the constant limit of zero.