Stability in the almost everywhere sense

A linear transfer operator approach

R. Rajaram, U. Vaidya, Makan Fardad, B. Ganapathysubramanian

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.

Original languageEnglish (US)
Pages (from-to)144-156
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume368
Issue number1
DOIs
StatePublished - Aug 2010

Fingerprint

Transfer Operator
Nonnegative Solution
Lyapunov
Linear Operator
Ordinary differential equation
Perron-Frobenius
Ordinary differential equations
Uniform Stability
Infinitesimal Generator
Linear partial differential equation
Advection
Global Stability
Stability Condition
Nonlinear Differential Equations
Numerical Methods
Convergence of numerical methods
Finite Element
Partial differential equations
Numerical methods
Differential equations

Keywords

  • Advection equation
  • Almost everywhere stability
  • Density function

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Stability in the almost everywhere sense : A linear transfer operator approach. / Rajaram, R.; Vaidya, U.; Fardad, Makan; Ganapathysubramanian, B.

In: Journal of Mathematical Analysis and Applications, Vol. 368, No. 1, 08.2010, p. 144-156.

Research output: Contribution to journalArticle

Rajaram, R. ; Vaidya, U. ; Fardad, Makan ; Ganapathysubramanian, B. / Stability in the almost everywhere sense : A linear transfer operator approach. In: Journal of Mathematical Analysis and Applications. 2010 ; Vol. 368, No. 1. pp. 144-156.
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