[Paper Review] A theoretical investigation of Brockett's ensemble optimal control problems

Jan Bartsch, Alfio Borzı̀+2 more|arXiv (Cornell University)|Sep 26, 2018

Stability and Controllability of Differential EquationsEngineering25 references18 citations

TL;DR

This paper establishes a rigorous theoretical framework for Brockett's ensemble optimal control problems governed by the Liouville equation with unbounded drift coefficients. It proves well-posedness in weighted Sobolev spaces, ensures existence and uniqueness of optimal controls, and derives optimality systems for non-smooth, constrained control problems involving $L^2$, $H^1$, and $L^1$ control costs.

ABSTRACT

This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework. Following Brockett's formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.

Motivation & Objective

• To develop a well-posedness theory for the Liouville equation with unbounded drift coefficients arising in ensemble control.
• To extend existence and uniqueness results to weighted Sobolev spaces to ensure continuity and Fréchet differentiability of the control-to-state map.
• To formulate and analyze non-smooth optimal control problems with diverse cost functionals, including $L^2$, $H^1$, and $L^1$ control costs.
• To incorporate box control constraints and characterize optimal controls via first-order optimality systems.
• To establish existence and uniqueness of optimal controls under suitable assumptions on the drift and cost functionals.

Proposed method

• Formulate ensemble optimal control problems governed by the Liouville equation with control-in-the-coefficients drift: $a(t,x;u,v) = \overline{a} + \overline{b}u(t) + \overline{c}v(t)x$.
• Apply the DiPerna-Lions theory to establish well-posedness in $H^m$ spaces for drifts with at most linear growth in $x$.
• Extend the well-posedness framework to weighted Sobolev spaces $H^m_k$ to handle unbounded coefficients and ensure regularity of the control-to-state map.
• Use approximation techniques involving mollifiers and time regularization to handle time-dependent drifts and densities.
• Derive first-order optimality systems by applying calculus of variations and adjoint methods to non-smooth cost functionals.
• Characterize optimal controls as solutions to coupled systems of forward (Liouville) and backward (adjoint) PDEs with appropriate boundary and initial conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the Liouville equation with unbounded drift coefficients admit a unique solution in weighted Sobolev spaces?
• RQ2 How can the control-to-state map be shown to be continuous and Fréchet differentiable in the context of ensemble control with unbounded coefficients?
• RQ3 What are the necessary and sufficient conditions for the existence and uniqueness of optimal controls in non-smooth ensemble optimal control problems?
• RQ4 How do different control cost functionals—$L^2$, $H^1$, and $L^1$—affect the structure and properties of the optimality system?
• RQ5 Can box constraints on the control be incorporated into the theoretical framework while preserving existence and uniqueness of optimal solutions?

Key findings

• Well-posedness of the Liouville equation is established in weighted Sobolev spaces $H^m_k$ for drifts with at most linear growth in $x$, under integrability and regularity conditions on the drift.
• The control-to-state map $G: (u,v) \mapsto \rho$ is shown to be continuous and Fréchet differentiable in the weighted Sobolev framework, enabling variational analysis.
• Existence of optimal controls is proven for non-smooth optimal control problems involving $L^2$, $H^1$, and $L^1$ control costs, as well as box constraints.
• Uniqueness of optimal controls is established under suitable convexity and regularity assumptions on the cost functional and drift.
• Optimal controls are characterized as solutions to a system of optimality equations consisting of the forward Liouville equation and an adjoint backward equation.
• The convergence of approximation sequences involving mollifiers and time regularization is rigorously proven, ensuring stability and consistency of the theoretical framework.