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Characterisation of Stability and Decay Rates in a Weakly Damped Second Order Linear Differential Equation

John A. D. Appleby, Subham Pal

Abstract

This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are asymptotically stable. Conditions are also given which characterise when the solution and its derivative tend to zero. Finally, we give sharp sufficient conditions under which the solution of the forced equation has the same asymptotic behaviour as the unforced equation, to leading order.

Characterisation of Stability and Decay Rates in a Weakly Damped Second Order Linear Differential Equation

Abstract

This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are asymptotically stable. Conditions are also given which characterise when the solution and its derivative tend to zero. Finally, we give sharp sufficient conditions under which the solution of the forced equation has the same asymptotic behaviour as the unforced equation, to leading order.

Paper Structure

This paper contains 16 sections, 30 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Transformation to the Unperturbed Form
  3. Integrability of the Perturbation
  4. Variation of Constants and Resolvents
  5. Asymptotic Estimates and Error Bounds
  6. Necessary Conditions for Asymptotic Decay
  7. Sufficient Conditions for Asymptotic Decay
  8. Sharp Asymptotic Kernel Estimates and Leading-Order Decay
  9. Second-Order Filtering: The Function $y_2(t)$
  10. Preservation of Unperturbed Dynamics of $x$
  11. Subexponential Decay Bounds
  12. Unperturbed dynamics preserved implies $y_2=o(1/A)$
  13. Representation of the Solution; Bounds on Kernels
  14. Decomposition of the Kernel $P_4(t, s)$
  15. Asymptotic behaviour of the integral $P_4\ast y_2$
  16. ...and 1 more sections

Key Result

Lemma 2

Define the function $u(t)$ via the transformation: Then $u(t)$ satisfies the differential equation: where the perturbation $q(t)$ and the new forcing $g(t)$ are given by:

Theorems & Definitions (62)

  • Definition 1: Hypotheses on the Coefficients
  • Lemma 2: Elimination of Damping
  • proof
  • Theorem 3: $L^1$ Integrability of $q$
  • proof
  • Theorem 4: Variation of Constants
  • proof
  • Theorem 5: Green's Function Expansion
  • proof
  • Lemma 6: Convolution of a Decaying Function
  • ...and 52 more