Optimal perturbations for nonlinear systems using graph-based optimal transport

Abstract

We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge-Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

Optimal transport over nonlinear systems via infinitesimal generators on graphs

“…This work extends our previous work [39] in several directions. First, we work in continuoustime, and as a result, the (passive) dynamics and the control act on the measure concurrently (rather than in a switching fashion).…”

“…It is then proved that arbitrary final measures not lying on the boundary of the probability simplex can be reached in finite-time. This work extends our previous work [39] in several directions. First, we work in continuoustime, and as a result, the (passive) dynamics and the control act on the measure concurrently (rather than in a switching fashion).…”

“…This also removes the requirement in Ref. [39] that the control acts on faster time scales than the passive dynamics. Second, rather than assuming full-controllability of measures, we rigorously relate the controllability in the space of measures on a graph to the controllability properties of the underlying (single-agent) dynamical system in continuous phase space.…”

“…In our previous work [39], we studied the problem of obtaining optimal perturbations in discrete time for systems with nonlinear dynamics, that result in desired measure transport. The perturbations were modeled as instantaneous, and computed by solving a Monge-Kantorovich optimal transport problem on a graph in pseudo-time.…”

Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

“…This rate of convergence is determined by the magnitude of the second eigenvalue λ 2 of the transfer operator P and we determine the perturbation that pushes the eigenvalue farthest from the unit circle. Related perturbative approaches include [13], where the mixing rate of (possibly periodically driven) fluid flows was increased by perturbing the advective part of the dynamics and solving a linear program; [14], where similar kernel perturbation ideas were used to drive a nonequilibrium density toward equilibrium by solving a convex quadratic program with linear constraints; and [18] where a governing flow is perturbed deterministically so as to evolve a specified initial density into a specified final density over a fixed time duration, with the perturbation determined as the numerical solution of a convex optimisation problem. In the current setting, our perturbation acts on the stochastic part of the dynamics and we can find a solution in closed form after some preliminary computations.…”