A theoretical investigation of Brockett’s ensemble optimal control problems

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.

The Pontryagin maximum principle for solving Fokker–Planck optimal control problems

“…For example as in [15,16], one could choose A x, t = x − x d (t) 2 , and similarly F x = x − x T 2 . Moreover, as in [8], we may…”

“…Our choice of a specific openloop control structure is motivated by the discussion in [15,16] in the framework of ensemble controls, where it is pointed out that a composite linear-bilinear openloop control mechanism in the SDE may provide a reasonable approximation of a closed-loop control; we investigate this fact with numerical experiments. We also refer to [15,16] and the recent work [8] concerning the choice of the functions G and F in the cost functional. These functions will be specified and discussed below.…”

“…The choice of A and F is motivated by the concept of ensemble control proposed in [15,16] and analysed in [8]. To illustrate this setting, suppose that the purpose of the control is that the trajectories of our stochastic models are required to track the The main purpose of our experiments is to validate the resulting optimal control problems at the level of the stochastic dynamics.…”

A numerical investigation of Brockett’s ensemble optimal control problems

“…The formulation and theoretical investigation of the ensemble control problems considered in this paper are presented in [3], where existence and regularity of solutions for a class of Liouville optimal control problems is discussed in the case where the drift function corresponds to a composition of linear and bilinear (input-affine) control mechanisms for the underlying dynamical system. In this class of problems, the purpose of the controls is to steer the ensemble of trajectories along a given path and to come close to a desired target configuration at a given final time.…”

“…As thoroughly discussed in [3] and illustrated below, the first step in solving our ensemble optimal control problems is the derivation of the corresponding first-order optimality conditions that consist of the controlled Liouville equation, its optimization adjoint (having the structure of a transport equation), and a variational inequality that we may also call (with some abuse of wording) the optimality condition equation. The numerical solution of this optimality system proceeds along two main steps that are the numerical approximation of the equations involved and their solution by a numerical optimization scheme.…”