Stability Of Nonautonomous Differential Equations Research Articles

Unstable Solutions of Nonautonomous Linear Differential Equations

The fact that the eigenvalues of the family of matrices $A(t)$ do not determine the stability of nonautonomous differential equations $x'=A(t)x$ is well known. This point is often illustrated using examples in which the matrices $A(t)$ have constant eigenvalues with negative real part, but the solutions of the corresponding differential equation grow in time. Here we provide an intuitive, geometric explanation of the idea that underlies these examples. The discussion is accompanied by a number of animations and easily modifiable Mathematica programs. We conclude with a discussion of possible extensions of the ideas that may provide suitable topics for undergraduate research.

Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v ′ = A ( t ) v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v ′ = A ( t ) v + f ( t , v ) under sufficiently small perturbations f. This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation v ′ = A ( t ) v is Lyapunov regular if for every k the limit of Γ ( t ) 1 / t as t → ∞ exists, where Γ ( t ) is any k-volume defined by solutions v 1 ( t ) , … , v k ( t ) . We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.

A New Criterion for Asymptotic Stability of Nonautonomous Differential Equations: An Illustrative Example

This paper discusses asymptotic stability for nonautonomous systems by means of the direct method of Liapunov. We consider a positive time-invariant Liapunov function with negative semi-definite derivative. We focus on the extra conditions needed in order to guarantee asymptotic stability and propose a new criterion. The theorem is illustrated with a simple example: we discuss the harmonic oscillator with time-variant damping for which we derive a new asymptotic stability result.

Semigroups and Stability of Nonautonomous Differential Equations in Banach Spaces

Semigroups and Stability of Nonautonomous Differential Equations in Banach Spaces

Semigroups and stability of nonautonomous differential equations in Banach spaces

This paper is concerned with nonautonomous differential equations in Banach spaces. Using the theory of semigroups of linear and nonlinear operators one investigates the semigroups of weighted translation operators associated with the underlying equations. Necessary and sufficient conditions for different types of stability are given in terms of spectral properties of the translation operators and the differential operators associated with the equations.

On the Stability of Nonautonomous Differential Equations $\dot x = [A + B(t)]x$, with Skew Symmetric Matrix $B(t)$

In this paper we characterize (in Theorem 1) the uniform asymptotic stability of equations of the form \[\left[ {\begin{array}{*{20}c} {\dot x} \\ {\dot y} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A(t)} &\vline & { - B(t)} \\\hline {B(t)} &\vline & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right]\] (where $A(t) + A(t)^T $ is negative definite) in terms of the “richness” of $B(t)$. The equation is uniformly asymptotically stable if and only if $B(t)$ is sufficiently rich. We actually obtain stability results for a much broader class of systems (Theorems 2 and 3) whose behavior is similar to the one above. Such systems have come up recently in some adaptive control problems.

Conditions for the stability of nonautonomous differential equations

Conditions for the stability of nonautonomous differential equations