Optimal transport over nonlinear systems via infinitesimal generators on graphs

Abstract

We present a set-oriented graph-based computational framework for continuoustime 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.2010 Mathematics Subject Classification. Primary: 93C10, 47D03, 37M99; Secondary: 93C20.

Data-Driven Approximations of Dynamical Systems Operators for Control

“…Optimal control approaches have also been proposed to modify the pdf in the presence of uncertainty in initial conditions or parameters [153,139]. The process of driving one distribution to another one is further intimately related to Monge-Kantorovich optimal transport theory [123,83,52,68]. In [68], optimal transport theory has been used to solve an finite-horizon control problem to achieve a desired distribution, where the optimal control vector field is estimated by solving the associated Liouville equation over the finite horizon.…”

“…In [68], optimal transport theory has been used to solve an finite-horizon control problem to achieve a desired distribution, where the optimal control vector field is estimated by solving the associated Liouville equation over the finite horizon. Set-oriented and graph-based methods have also been used to study controllability and optimal transport [53].…”

Optimal Transport Over Deterministic Discrete-Time Nonlinear Systems Using Stochastic Feedback Laws

“…We consider the system The final time is set to N = 10. To define the map F : X → X, we consider the double-gyre system [9]: The set X is not invariant for all choices of control inputs in U . Hence, since this set must be approximatable by a finite set, we define F (x) + G(u) x if F (x) + G(u) / ∈ X for some (x, u) ∈ X × U .…”

“…There have also been efforts to extend the theory to nonlinear driftless control-affine systems in the framework of sub-Riemannian optimal transport [1], [10], [14]. See also [9], in which we develop connections between computational optimal transport over continuoustime nonlinear control systems and optimal transport on finite state spaces. Closely related to such optimal transport problems is the theory of mean-field games and mean-field type controls [2], [7], [18].…”

“…Next, we show that we can frame the control-constrained optimal transport problem of controllable nonlinear systems as a linear programming problem, as in the Kantorovich formulation of the optimal transport problem. Unlike our previous work [9], which focused on computational aspects of optimal transport problems for nonlinear systems with a particular control-affine structure, in this paper we solve the optimal transport problem for general nonlinear control systems in discrete time.…”