In this paper, a basic property that a Liapunov-Krasovskii functional must have when used for studying the stability of time delay systems under Carath¶eodory conditions is studied. This property concerns the equality, almost everywhere, of the upper right-hand Dini derivative of the Liapunov-Krasovskii functional evaluated at the solution, and of the Liapunov-Krasovskii derivative (computed by...

We present a sufficient condition for robust permanence of ecological (or Kolmogorov) differential equations based on average Liapunov functions. Via the minimax theorem we rederive Schreiber’s sufficient condition [S. Schreiber, J. Differential Equations, 162 (2000), pp. 400–426] in terms of Liapunov exponents and give various generalizations. Then we study robustness of permanence criteria ag...

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré–Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are al...

Dynamical systems with several equilibria occur in various fields of science and engineering: electrical machines, chemical reactions, economics, biology, neural networks. As pointed out by many researchers, good results on qualitative behaviour of such systems may be obtained if a Liapunov function is available. Fortunately for almost all systems cited above the Liapunov function is associated...

In this paper we propose a finite element discretization of the Maxwell-Landau-Lifchitz-Gilbert equations governing the electromagnetic field in a ferromagnetic material. Our point of view is that it is desirable for the discrete problem to possess conservation properties similar to the continuous system. We first prove the existence of a new class of Liapunov functions for the continuous probl...

This paper provides a converse Liapunov theorem for uniformly locally exponentially stable, locally Lipschitz, non-linear, time-varying, possibly non-smooth systems that admit Carathéodory solutions. The main result proves that a critical point of such a system is uniformly locally exponentially stable if and only if the system admits a local (possibly non-smooth, timevarying) Liapunov function.

In this paper, the concepts and the direct theorems of stability in the sense of Liapunov, within the framework of Birkhoffian dynamical systems on manifolds, are considered. The Liapunov-type functions are constructed for linear and nonlinear LC and RLC electrical networks, to prove stability under certain conditions.

In this paper a generalization of avoidance control for multi-agent dynamic systems is presented. Strategies for avoidance control for multiple agents are obtained using individual Liapunov-type functions. The overall system avoidance conditions are guaranteed using a vector Liapunov-type function via an M-matrix property.

This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. The existence of a positive time-invariant Liapunov function with negative semi-definite derivative is assumed. The paper focuses on the extra conditions needed in order to guarantee asymptotic stability. The proposed criterion is compared with the standard results available for autono...

In [40], it was shown that the following singularly perturbed Dirichlet problem 2∆u− u+ |u|p−1u = 0, in Ω, u = 0 on ∂Ω has a nodal solution u which has the least energy among all nodal solutions. Moreover, it is shown that u has exactly one local maximum point P 1 with a positive value and one local minimum point P 2 with a negative value and, as → 0, φ(P 1 , P 2 ) → max (P1,P2)∈Ω×Ω φ(P1, P2), ...