New Conditions for the Exponential Stability of Nonlinear Differential Equations

Abstract

We develop a method for proving local exponential stability of nonlinear nonautonomous differential equations as well as pseudolinear differential systems. The logarithmic norm technique combined with the "freezing" method is used to study stability of differential systems with slowly varying coefficients and nonlinear perturbations. Testable conditions for local exponential stability of pseudo-linear differential systems are given. Besides, we establish the robustness of the exponential stability in finite-dimensional spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. We illustrate the application of this test to linear approximations of the differential systems under consideration.

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

“…The stability of nonautonomous perturbed dynamical systems has attracted the attention of many researchers and has produced several important results. [1][2][3][4][5] It turns out that time-varying differential equations appear as a natural description of observed evolution phenomena of various real-world problems where the study of asymptotic stability is more interesting than stability. Indeed, the study of asymptotic stability of dynamical systems is one of the most important research area in system design.…”