### 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) |
---|---|

Article number | 8410425 |

Pages (from-to) | 168-173 |

Number of pages | 6 |

Journal | IEEE Control Systems Letters |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Control and Optimization

### Cite this

*IEEE Control Systems Letters*,

*3*(1), 168-173. [8410425]. https://doi.org/10.1109/LCSYS.2018.2855185

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

Research output: Contribution to journal › Article

*IEEE Control Systems Letters*, vol. 3, no. 1, 8410425, pp. 168-173. https://doi.org/10.1109/LCSYS.2018.2855185

}

TY - JOUR

T1 - Optimal Transport over Deterministic Discrete-Time Nonlinear Systems Using Stochastic Feedback Laws

AU - Elamvazhuthi, Karthik

AU - Grover, Piyush

AU - Berman, Spring

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85057637907&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057637907&partnerID=8YFLogxK

U2 - 10.1109/LCSYS.2018.2855185

DO - 10.1109/LCSYS.2018.2855185

M3 - Article

VL - 3

SP - 168

EP - 173

JO - IEEE Control Systems Letters

JF - IEEE Control Systems Letters

SN - 2475-1456

IS - 1

M1 - 8410425

ER -