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Center for Nonlinear Studies |
For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Bounds can be proved a priori using so-called auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding extremal trajectories and optimal auxiliary functions are strongly dual and thus that this approach provides arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where extremal trajectories must lie. Moreover, for polynomial systems, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382-386 (2018).
Host: Yen Ting Lin and Angel Garcia