Mean-field sparse optimal control

Abstract

We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modelling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. In the classical mean-field theory, one studies the behaviour of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect. In this paper, we address instead the situation where the leaders are actually influenced also by an external policy maker , and we propagate its effect for the number N of followers going to infinity. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Γ -limit of the finite dimensional sparse optimal control problems.

Sparse Jurdjevic–Quinn stabilization of dissipative systems

“…With the same method, in [5] a non-global sparse stabilization method was proved for a system submitted to repulsion and attraction forces (see [16]). Beside the sparse controllability, we mention also the controllability via leadership, which deals with single-input control-affine systems (or when m n, see [1,2,7,20,33,17]. We consider a first-order model for N agents, represented by the vector of their positions x ∈ R N , interacting one with each other according tȯ…”

“…It is therefore natural to seek controls achieving the same goal with less active components. This is the case for instance when we want only one leader to act on a whole crowd (such as a dog with a flock of sheep), or more generally when feasible control strategies are required to focus on a small number of agents at each time (see [1,2,6,7,20,33]). …”

Mean-field optimal control by leaders

“…Another direction of investigation is the study of the limiting behavior when the number of agents tends to infinity, see [14], [13], [6]. The recent work [12] combined the two issues of control strategies and meanfield limit, by providing convergence results based on the concept of Γ-limit. Other authors addressed the problem of controlling a whole group by a limited number of leaders mostly at a microscopic level, see for instance [18], [4], [11], [15], [16] and references therein.…”

Partial differential equation models in the socio-economic sciences

“…The differences from mean-field games arising in such are further discussed in [54]. In Fornasier et al [55], a particular instance of mean-field optimal control by sparse controls is discussed. The major idea is to find optimal controls for such systems with as few points to act as possible, marrying ideas from mean-field control with recent advances on sparsity in imaging and compressed sensing.…”