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Optimal Control in the Wasserstein Space: How to Control a Distribution?
This thesis aims at laying the foundations for optimal control in the Wasserstein space (i.e., the space of probability distributions endowed with the Wasserstein distance).
Keywords: Optimal transport, optimal control, Wasserstein space, Distribution steering, control theory
**Introduction**
Traditionally, optimal control deals with finding the best control for a dynamical system with respect to a given cost functional. In this project instead, we aim at optimally controlling the distribution related to a dynamical system. This problem has many control applications. First, it naturally deals with systems inherently described by a distribution (rather than a vector of real values), such as large swarms of (indistinguishable) robots. Second, it may serve as a control framework for stochastic systems, whereby the distribution is induced by the uncertainty affecting the system. Third, applications can be envisioned in the field of the control of small scale systems.
**Project Description**
This thesis aims at laying the foundations for optimal control in the Wasserstein space (i.e., the space of probability distributions endowed with the Wasserstein distance). Specifically, we would like to develop control methodologies to systematically synthesize feedback control laws for (discrete-time) dynamical systems described by a distribution. Starting from the generalization of the dynamic programming principle to the Wasserstein space, we plan to progressively investigate the effect of simplifying assumptions (e.g., linear dynamics, discrete and Gaussian distributions) and analytically characterize (and, if possible, compute) the related optimal control policy. If time permits, we are further interested in implementing the developed methodology on toy examples and in comparing it with existing control methodologies, such as stochastic and robust control.
**Introduction**
Traditionally, optimal control deals with finding the best control for a dynamical system with respect to a given cost functional. In this project instead, we aim at optimally controlling the distribution related to a dynamical system. This problem has many control applications. First, it naturally deals with systems inherently described by a distribution (rather than a vector of real values), such as large swarms of (indistinguishable) robots. Second, it may serve as a control framework for stochastic systems, whereby the distribution is induced by the uncertainty affecting the system. Third, applications can be envisioned in the field of the control of small scale systems.
**Project Description**
This thesis aims at laying the foundations for optimal control in the Wasserstein space (i.e., the space of probability distributions endowed with the Wasserstein distance). Specifically, we would like to develop control methodologies to systematically synthesize feedback control laws for (discrete-time) dynamical systems described by a distribution. Starting from the generalization of the dynamic programming principle to the Wasserstein space, we plan to progressively investigate the effect of simplifying assumptions (e.g., linear dynamics, discrete and Gaussian distributions) and analytically characterize (and, if possible, compute) the related optimal control policy. If time permits, we are further interested in implementing the developed methodology on toy examples and in comparing it with existing control methodologies, such as stochastic and robust control.
The student will:
(1) learn and understand concepts related to the topic, including optimal control theory, optimal transport, and the Wasserstein distance/space;
(2) explore and review the relevant literature;
(3) derive control methods to optimally control a distribution;
(4) implement the developed methodology and compare it to traditional control strategies (e.g., stochastic control and robust control).
Of course, the project can be adapted on the run if new interesting research directions arise. Moreover, if the results are promising they can be turned into a publication.
**Prerequisites**
The project is well suited for a student who enjoys mathematics. A background in analysis, linear algebra, and optimal control is required. Some basic knowledge of probability theory/stochastics is beneficial.
The student will: (1) learn and understand concepts related to the topic, including optimal control theory, optimal transport, and the Wasserstein distance/space; (2) explore and review the relevant literature; (3) derive control methods to optimally control a distribution; (4) implement the developed methodology and compare it to traditional control strategies (e.g., stochastic control and robust control). Of course, the project can be adapted on the run if new interesting research directions arise. Moreover, if the results are promising they can be turned into a publication.
**Prerequisites**
The project is well suited for a student who enjoys mathematics. A background in analysis, linear algebra, and optimal control is required. Some basic knowledge of probability theory/stochastics is beneficial.
Do not hesitate to contact me if you are interested in the project or, more broadly, in the research topics of optimal transport, Wasserstein space, and their applications.
Nicolas Lanzetti, lnicolas@ethz.ch
Do not hesitate to contact me if you are interested in the project or, more broadly, in the research topics of optimal transport, Wasserstein space, and their applications.