Search results
Results from the WOW.Com Content Network
Dirac delta function; ... Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay. Exponential stability is a ...
The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations.
Other names for linear stability include exponential stability or stability in terms of first approximation. [ 1 ] [ 2 ] If there exists an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
The exponential function becomes arbitrarily steep as x → ∞, and therefore is not globally Lipschitz continuous, despite being an analytic function. The function f(x) = x 2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.
Julia provides package StableDistributions.jl which has methods of generation, fitting, probability density, cumulative distribution function, characteristic and moment generating functions, quantile and related functions, convolution and affine transformations of stable distributions. It uses modernised algorithms improved by John P. Nolan.
Exponential backoff is an algorithm that uses feedback to multiplicatively decrease the rate of some process, in order to gradually find an acceptable rate. These algorithms find usage in a wide range of systems and processes, with radio networks and computer networks being particularly notable.