The structure of the level surfaces of a Lyapunov function

Abstract

Let f be a real-valued Cl function which is defined on Euclidian space R". We are interested in characterizing the noncritical level surfaces off near its isolated relative maxima and minima. The technique which is used for this investigation is to study the relationship between the trajectories of a differential equation and its Lyapunov function. As an application of interest, we obtain characterizations of the level surfaces of a Lyapunov function and of the domain of asymptotic stability of an asymptotically stable critical point. The domain of asymptotic stability is diffeomorphic to R", and the

Growth transformations for functions on manifolds

“…In Theorems 2 and 3 one is able to assert that the neighborhood V is a disk provided the function P is a C 2 -function, cf. [4]. Also,, these results are really " local" results, so they are still valid even if σ and P are defined only in a neighborhood of the maximum q.…”

Homogeneous Lyapunov functions and necessary conditions for stabilization

“…Adding a dimension, the (n + 1)-dimensional system In contrast to the first example, the refined index condition of Theorem 2 is not satisfied. In other words, it does not help in the present example to consider another manifold in order to satisfy condition (22): it is easily verified on Fig. 1 that not only every continuous function v(O) defined in (-o% +oe) but also every continuous path defined in the plane (0, v) for 0 e (-o% +oe) either lies in the region a2(x, u)> 0 or intersects the region az(X, u)= 0.…”

“…Moreover, the manifold M* can be chosen to be the graph of an explicit function of the original coordinates. On the contrary, condition (22) does not imply condition (21). In particular, the manifold M* might be the graph of an implicit function h(x, y) = 0.…”

“…In general, the level sets of a Lyapunov function are merely homotopy spheres [22], hence (c) cannot be proven because of the lack of a proof of the Poincar6 conjecture. Property (b) of Proposition 1, usually called the Euler property, emphasizes that the vector field F nowhere points outward "radially," i.e., that, for x ~ 0, the equality…”

A Generalization of Zubov's Method to Perturbed Systems

“…the analysis of power systems [1] and turbulence phenomena in fluid dynamics [3,9,19]. Several papers and books discuss theoretical [21,22,7,12] as well as computational aspects [20,13,1,10] of this problem.…”