Coherent Structures and Optimal Control Theory

Abstract

Coherent structures are persistent, large-scale spatiotemporal features in fluid flow fields. Optimal control theory is a branch of mathematics highly relevant to engineering for manipulating the behavior of dynamical systems. In this work, we explore the interplay and connections that arise between these two concepts in two different contexts. In the first context, we are motivated by the problem of path planning of ocean drifters moving within flow fields. Here, we are interested in how the background coherent structures characterized by the Lyapunov exponents of the flow impact and shape energy efficient trajectories of the robot moving within. In the second context, we investigate a strategy for controlling the dynamics of swirling vortex structures in flow fields. In this case, we consider “coherent structures” of the flow field characterized by Koopman eigenfunctions. In particular, we investigate how formulating control in terms of these eigenfunctions changes the vortex dynamics. Given that there has been much recent work in data-driven computation of these coherent structures, we anticipate this work to be of considerable interest to scientists and engineers.