Mean-Field Selective Optimal Control via Transient Leadership

Abstract

A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. In this way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via $$\varGamma $$ Γ -convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments.

Optimal control problems in transport dynamics with additive noise

“…Thanks to assumptions (φ1) and (φ2), to each term in the sum on the right-hand side of (4.22) we can apply [6, Lemma 9.4.3] (see also [1,Proposition 5]). Thus, for j = 1, .…”

“…The presence of the term g(µ t ) is an additional modeling possibility allowing the policy maker to tune the control action on the actual state of the system. The above class of control vector fields is, in principle, the one considered in [1,8,26,34,42]. In [8,42], however, well posedness of the optimal control problems was considerably simplified by the assumption that the leaders' population remain discrete.…”

“…In our setting, we have instead to resort to the superposition principle [5,Theorem 5.2], [6,Chapter 8], [7], as it will be clear from the proof of Theorem 4.6 below. This tool has already proven to be crucial in connection with the problems considered in [1,21,26,34], where however no diffusive terms were present in the state equations. In our setting its use has therefore to be combined with some a priori estimates for equation (1.1) which are recovered by looking at its stochastic Lagrangian counterpart (1.2) and employing some fixed point argument.…”

“…The latter is a rather effective tool to overcome the curse of dimensionality for systems with a very large number of agents. Indeed, kinetic approximations of multi-agent systems and mean-field optimal control problems, mostly in the deterministic setting, have been proposed in recent literature in connection with a huge number of possible applications, ranging from models for opinion formation [30,51], wealth distribution [29,31,43], traffic or pedestrian flows [2,28,46,47,52], herding problems [1,2,9,20,40,48,54], consensusbased optimization [17,24,33,53] (see also [18,26,38,44,45] for rigorous derivations and further applications and [13,14,15,16,19] for optimality conditions). In the context of multi-agent systems with stochastic noise, but without control, we also refer the reader to [11], while mean-field control problems with diffusion terms have been recently considered in [3,25].…”