In engineering practice, damping control is a usual method for stabilization. In this paper we consider nonlinear affine systems. We show that if there exists a feedback law which provides sufficiently fast convergence of trajectories (rational stabilization) then the system can also be stabilized by damping control. Our construction relies on a Liapunov function, which is basically defined as ...

The chemical implementation of a neuron and connections among neurons described in prior work is used to construct collective neural networks. With stated approximations, these chemical networks are reduced to networks of the Hopfield type. Chemical networks approaching a stationary or equilibrium state provide a Liapunov function with the same extremal properties as Hopfield's energy function....

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...

We show that the feedback from the macroscopic dynamics of a system of coupled units can synchronize the dynamics of these units. We studied the dynamics of maps coupled through their variables and control parameters. The feedback adjusted the values of the parameters of each map by using a function that depended on the difference between the Liapunov exponent of each unit and the Liapunov expo...

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...

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...