Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
Abstract
For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) longtime average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.
 Publication:

Physics Letters A
 Pub Date:
 February 2018
 DOI:
 10.1016/j.physleta.2017.12.023
 arXiv:
 arXiv:1705.07096
 Bibcode:
 2018PhLA..382..382T
 Keywords:

 Nonlinear dynamical systems;
 Time averages;
 Ergodic optimization;
 Semidefinite programming;
 Sumofsquares polynomials;
 Lorenz equations;
 Mathematics  Dynamical Systems;
 Mathematics  Optimization and Control;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 Minor revisions from review