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

Karthik Elamvazhuthi, Piyush Grover, Spring Berman

Research output: Contribution to journalArticle

Abstract

This letter considers the relaxed version of the transport problem for general nonlinear control systems, where the objective is to design time-varying feedback laws that transport a given initial probability measure to a target probability measure under the action of the closed-loop system. To make the problem analytically tractable, we consider control laws that are stochastic, i.e., the control laws are maps from the state space of the control system to the space of probability measures on the set of admissible control inputs. Under some controllability assumptions on the control system as defined on the state space, we show that the transport problem, considered as a controllability problem for the lifted control system on the space of probability measures, is well-posed for a large class of initial and target measures. We use this to prove the well-posedness of a fixed-endpoint optimal control problem defined on the space of probability measures, where along with the terminal constraints, the goal is to optimize an objective functional along the trajectory of the control system. This optimization problem can be posed as an infinite-dimensional linear programming problem. This formulation facilitates numerical solutions of the transport problem for low-dimensional control systems, as we show in two numerical examples.

Original language English (US) 8410425 168-173 6 IEEE Control Systems Letters 3 1 https://doi.org/10.1109/LCSYS.2018.2855185 Published - Jan 1 2019

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Optimal Transport
Nonlinear Stochastic Systems
Nonlinear Discrete-time Systems
Feedback Law
Nonlinear systems
Probability Measure
Feedback
Control systems
Control System
Controllability
State Space
Nonlinear control systems
Nonlinear Control Systems
Target
Closed loop systems
Linear programming
Well-posedness
Closed-loop System
Trajectories
Optimal Control Problem

ASJC Scopus subject areas

• Control and Systems Engineering
• Control and Optimization

Cite this

Optimal Transport over Deterministic Discrete-Time Nonlinear Systems Using Stochastic Feedback Laws. / Elamvazhuthi, Karthik; Grover, Piyush; Berman, Spring.

In: IEEE Control Systems Letters, Vol. 3, No. 1, 8410425, 01.01.2019, p. 168-173.

Research output: Contribution to journalArticle

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