Practical Uniform Stability of Nonlinear Differential Delay Equations

Abstract

In this paper, we investigate the problem of global uniform practical exponential stability of a general nonlinear non autonomous differential delay equations. Using the global uniform practical exponential stability of the corresponding differential equation without delay, we show that the differential delay equation will remain globally uniformly practically exponentially stable provided that the time-lag is small enough. Finally, some illustrative examples are given to demonstrate the validity of the results.Mathematics Subject Classification (2010). Primary 93Dxx; Secondary 34Dxx.

Stabilization of a certain class of fuzzy control systems with uncertainties

“…As a first step, we need to recall what is meant by uniformly ultimately bounded and uniform global practical exponential stability of dynamic systems ( [2], [4], [5]). Consider a system described byẋ…”

Practical exponential stability in mean square of stochastic partial differential equations

“…We state now the definitions of the almost surely convergence of solutions to a small closed ball B r ⊂ H centered at zero with radius r (see [1]- [6], [10]), and we will consider initial values in the space X 0 ∈ L 2 (Ω, F 0 , P; H).…”

Practical Stability of Stochastic Delay Evolution Equations

“…It is therefore of great interest to be able to characterize the behavior of the solutions. Several interesting and important variants to Lyapunov's original concepts of stability were proposed in [1]- [3] and [14]. However, when the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions with respect to a small neighborhood of the origin, what yields to the concept of practical stability.…”

“…However, when the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions with respect to a small neighborhood of the origin, what yields to the concept of practical stability. It is worth mentioning some previous works on practical stability in the deterministic framework, as for example, [1], [3]. Also we would like to mention here the references [4,5,19,20], among others.…”