Calculus of Variations and Geometric Measure Theory

A. Ponosov - E. Stepanov

Smooth dynamics becomes hybrid in the limit

created by stepanov on 25 Feb 2016
modified on 13 Jan 2017


Published Paper

Inserted: 25 feb 2016
Last Updated: 13 jan 2017

Journal: SIAM J. Control and Optimization
Volume: 55
Number: 1
Pages: 29-50
Year: 2017
Doi: 10.1137/16M1064611


We show the appearance of an essentially nonlocal dynamics describing the limit behavior of trajectories of a class of dynamical systems defined by classical autonomous ODEs with smooth right-hand sides containing a small parameter and becoming discontinuous in the formal limit. The limit dynamics is shown to be described by an explicitly constructed Nerode-Kohn \emph{hybrid} dynamical system consisting of a continuous plant (ODE) and a finite state machine which are interacting and producing hybrid dynamics with possible memory effects. We remark however that, from the ``statistical'' point of view, the limit behavior of an ensemble of trajectories can still be described by an ODE, with possibly time-dependent and discontinuous right-hand side depending on the chosen ensemble.