Mass Transportation with LQ Cost Functions

Hindawi, A.; Pomet, Jean‐Baptiste; Rifford, Ludovic

doi:10.1007/s10440-010-9595-1

Cited by 16 publications

(10 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the special case of linear dynamical systems with quadratic cost, mirroring the optimal control case, further analytical development and computational simplification has been made [43,16]. As described in Ref.…”

Section: 1mentioning

confidence: 99%

“…The field of optimal mass transportation [80] is concerned with optimal mapping of measures in different settings, including on graphs [54], and has deep connections with phase space transport in dynamical systems [26,8,5]. The problem of optimal transport in linear dynamical systems has been studied recently [43,16,15], resulting in several theoretical and computational advances.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Optimal transport over nonlinear systems via infinitesimal generators on graphs

Elamvazhuthi¹,

Grover²

2018

Journal of Computational Dynamics

View full text Add to dashboard Cite

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.

show abstract

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

Optimal transport over nonlinear systems via infinitesimal generators on graphs

Elamvazhuthi¹,

Grover²

2018

Journal of Computational Dynamics

View full text Add to dashboard Cite

show abstract

“…Such techniques were developed in [20], which dealt with optimal transport with nonholonomic constraints. In a similar problem configuration, the existence and uniqueness of transport maps were determined for linear-quadratic costs by [21]. Other works include distributed optimal transport for swarms of single-integrators [22], [23], Perron-Frobenius operator methods for computing optimal transport over nonlinear systems [24], and covariance control [25]- [27].…”

Section: Introductionmentioning

confidence: 99%

Discrete-Time Linear-Quadratic Regulation via Optimal Transport

Badyn

Miehling

Janak

et al. 2021

2021 60th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

In this paper, we consider a discrete-time stochastic control problem with uncertain initial and target states. We first discuss the connection between optimal transport and stochastic control problems of this form. Next, we formulate a linear-quadratic regulator problem where the initial and terminal states are distributed according to specified probability densities. A closed-form solution for the optimal transport map in the case of linear-time varying systems is derived, along with an algorithm for computing the optimal map. Two numerical examples pertaining to swarm deployment demonstrate the practical applicability of the model, and performance of the numerical method.

show abstract

“…More recently there has been a large interest in control and estimation of densities, and one key result is that certain density control problems of first-order integrators can be seen as optimal transport problems [4]. This correspondence can be extended to general dynamics, and thus the optimal transport problem can be interpreted as a density control problem of agents (subsystems) with general dynamics [10], [14], [38].…”

Section: Introductionmentioning

confidence: 99%

Graph-structured tensor optimization for nonlinear density control and mean field games

Ringh¹,

Haasler²,

Chen³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems, which can be seen as a generalization of multi-marginal optimal transport problems with graph-structured costs, appear in many applications. In particular, we show that it can be used to model and solve nonlinear density control problems, including convex dynamic network flow problems and multi-species potential mean field games. The method is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures we derive efficient methods for computing these projections. In particular, these graph structures are the ones that occur in convex dynamic network flow problems and multi-species potential mean field games. We also illustrate the methodology on numerical examples from these problem classes.

show abstract

Mass Transportation with LQ Cost Functions

Cited by 16 publications

References 14 publications

Optimal transport over nonlinear systems via infinitesimal generators on graphs

Optimal transport over nonlinear systems via infinitesimal generators on graphs

Discrete-Time Linear-Quadratic Regulation via Optimal Transport

Graph-structured tensor optimization for nonlinear density control and mean field games

Contact Info

Product

Resources

About