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Hyers-Ulam stability for finite-dimensional nonautonomous dynamics

Davor Dragičević

Abstract

The main purpose of this paper is to obtain necessary and sufficient conditions under which a nonautonomous, finite-dimensional and two-sided dynamics generated by a sequence of matrices or a linear ODE exhibits Hyers-Ulam stability. Specifically, in the case of discrete time we consider a nonautonomous difference equation with possibly noninvertible coefficients, while in the case of continuous time we deal with a nonautonomous ordinary differential equation without any bounded growth assumptions.

Hyers-Ulam stability for finite-dimensional nonautonomous dynamics

Abstract

The main purpose of this paper is to obtain necessary and sufficient conditions under which a nonautonomous, finite-dimensional and two-sided dynamics generated by a sequence of matrices or a linear ODE exhibits Hyers-Ulam stability. Specifically, in the case of discrete time we consider a nonautonomous difference equation with possibly noninvertible coefficients, while in the case of continuous time we deal with a nonautonomous ordinary differential equation without any bounded growth assumptions.
Paper Structure (4 sections, 13 theorems, 63 equations)

This paper contains 4 sections, 13 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Discrete time case
  3. Preliminaries
  4. Continuous time case

Key Result

Theorem 1

Suppose that the only bounded sequence $(x_n)_{n\le 0} \subset \mathbb R^d$ such that $x_0=0$ and $x_{n+1}=A_n x_n$, $n\le -1$ is the zero-sequence, i.e. $x_n=0$ for $n\le 0$. Then, the following properties are equivalent:

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Theorem 1
  • proof
  • Lemma 1
  • proof : Proof of the lemma
  • ...and 31 more