Optimal control problems in transport dynamics

Abstract

In the present paper we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones.

Leader formation with mean-field birth and death models

“…Moreover, s t is continuous with respect to time, with uniformly bounded mass due to (H4) and with uniformly bounded support due to (H5). Then, hypotheses of Lemma 2.9 are satisfied, hence µ F t is the unique solution of (55) and it satisfies the Duhamel's formula (12). It is clear that the previous properties hold for µ L too, with the same vector field w t and source −s t .…”

“…Such approach has the advantage of reducing the computational complexity of the models (overcoming the curse of dimensionality [10]) and allows the so-called microfundation of macromodels, i.e., the validation of the macroscopic dynamics from the coherence with the behavior of individuals (a central issue in the field of macroeconomics). The mean-field limit of systems of interacting agents has been thoroughly studied also in conjunction with irregular interaction kernel [17,32], control problems [3,15,29,30,38] and multiple populations [4,5,12,21]. Also models where the total mass of the system is not preserved in time, due to the presence of source (or sink) terms, have been considered (see for instance [45,).…”

A measure theoretic approach to traffic flow optimisation on networks

“…Indeed some experiments (see Work [18]) have shown that it is possible to avoid stop-and-go phenomena regulating the interactions among drivers by means of external agents (autonomous vehicles, traffic light, signaling panels,etc.). The approach in this section is inspired to [4] where the authors present an optimization problem for a transport equation in the euclidean space with the control represented by a second distribution µ evolving according to another transport equation. The dynamics of the autonomous cars is similar to the ones of rest of the driver, with the difference that it can be controlled in order to minimize the objective functional.…”

“…In [4,10,11], the authors consider optimal control problems for measure transport equations in the Euclidean space. Relying on a similar approach, we consider a model where, besides the driver distributions, the velocity field depends also on a external distribution which interacts with the original population in order to optimize, for example, traffic volume or average speed on the road network.…”

Instantaneous control of interacting particle systems in the mean-field limit

“…Recently, many researchers have investigated related problems: in [1] an evacuation problem using invisible agents in a crowd was studied and a similar scenario with visible agents in [39], [37] considers a Fokker-Planck feedback control strategy for crowd motion and [23], where optimal strategies for driving a mobile agent in a guidance by repulsion model are studied. Optimal control problems for transport processes are discussed, e.g., in [6]. An extensive overview on the actual mathematical approaches for behavioral social systems can be found in [4].…”