Optimal transport over nonlinear systems via infinitesimal generators on graphs

We present a set-oriented graph-based framework for continuous-time optimal transport over nonlinear dynamical systems. Our approach allows us to 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 nonlinear control-affine systems. The action of the controlled vector fields is approximated by a continuous-time flow on a graph obtained by discretizing the phase space. The edge weights of this graph are obtained by computing infinitesimal generators of the Perron-Frobenius operator of the control vector fields. The problem is reduced to a modified Monge-Kantorovich optimal transport on this graph, motivated by the Brenier-Benamou fluid dynamics formulation. The well-posedness of the optimal transport problem on graph is related to controllability of the underlying dynamical system. The resulting convex problem is solved via state-of-the-art solvers to global optimality. Using our computational framework, we study optimal transport in dynamical systems arising in chaotic fluid dynamics and nonholonomic vehicle dynamics. 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. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and also opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.