Abstract
This technical note addresses the stability analysis of nonlinear dynamic systems. Three main contributions are made. First, we show that the standard assumption of a continuous Lyapunov function can be (and in some cases must be) relaxed. We introduce the concept of the `weak' Lyapunov function, which requires that an annulus condition be satisfied. We believe that this annulus condition is a more natural construct, because it is precisely what is needed to make the forward Lyapunov theorem true. Second, we provide an example of a nonlinear system with stable equilibrium point that cannot be shown to be stable with a continuous Lyapunov function. Finally, we demonstrate a simpler and less restrictive proof of the converse Lyapunov theorem.
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