TY - JOUR

T1 - Stability in the almost everywhere sense

T2 - A linear transfer operator approach

AU - Rajaram, R.

AU - Vaidya, U.

AU - Fardad, M.

AU - Ganapathysubramanian, B.

PY - 2010/8

Y1 - 2010/8

N2 - 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.

AB - 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.

KW - Advection equation

KW - Almost everywhere stability

KW - Density function

UR - http://www.scopus.com/inward/record.url?scp=77951207305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951207305&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2010.02.032

DO - 10.1016/j.jmaa.2010.02.032

M3 - Article

AN - SCOPUS:77951207305

SN - 0022-247X

VL - 368

SP - 144

EP - 156

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -