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For any quantity of interest in a system governed by ordinary differentialequations, it is natural to seek the largest (or smallest) long-time averageamong 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 theproblems of finding maximal trajectories and minimal auxiliary functions arestrongly dual. Thus, auxiliary functions provide arbitrarily sharp upper boundson time averages. Moreover, any nearly minimal auxiliary function providesphase space volumes in which all nearly maximal trajectories are guaranteed tolie. For polynomial equations, auxiliary functions can be constructed bysemidefinite programming, which we illustrate using the Lorenz system.

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems