Optimal transport over nonlinear systems via infinitesimal generators on graphs
For researchers in control theory and multi-agent systems, this work provides a novel computational approach to optimal transport in nonlinear dynamics, though it is an incremental extension of existing graph-based and optimal transport methods.
This paper introduces a graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems, reducing the problem to a convex Benamou-Brenier type formulation on a graph. The framework yields provably optimal control laws for steering distributions in finite time, with applications to swarm control in chaotic and non-holonomic systems.
We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.