Optimal synchronization problem for a multi-agent system

Abstract

In this paper we investigate a time-optimal control problem in the space of positive and finite Borel measures on R d , motivated by applications in multi-agent systems. We provide a definition of admissible trajectory in the space of Borel measures in a particular non-isolated context, inspired by the so called optimal logistic problem, where the aim is to assign an initial amount of resources to a mass of agents, depending only on their initial position, in such a way that they can reach the given target with this minimum amount of supplies. We provide some approximation results connecting the microscopical description with the macroscopical one in the mass-preserving setting, we construct an optimal trajectory in the non isolated case and finally we are able to provide a Dynamic Programming Principle.

Minimal time problem for discrete crowd models with a localized vector field

“…[18]. Instead, more recent developments in the non-linear case (based on generalization of differential inclusions) have been recently described in [14,15,16].…”

“…The sequence {t 1 σn(1) , ..., t 1 σn(n) } is then decreasing and σ n is a minimizing permutation in (15). We deduce that M e (X 0 , X 1 ) = M e (X 0 , X 1 ).…”

“…Proof of Theorem 1.2. Consider M e (X 0 , X 1 ) given in (15). By relabeling particles, we assume that the sequence {t 0 i } i∈{1,...,n} is increasingly ordered.…”

“…By relabeling particles, we assume that the sequence {t 0 i } i∈{1,...,n} is increasingly ordered. Let σ 0 be a minimizing permutation in (15). We build recursively a sequence of permutations {σ 1 , ..., σ n } as follows:…”

Generalized dynamic programming principle and sparse mean-field control problems

“…Alternatively, keeping the same definition of c f , we can give another definition of the cost c leading to a value function that is an averaged minimum time function. This is inspired by [19], where the authors provide also a possible example of application in the case with no interactions. Let S ⊆ R d be non empty and closed, andS = {µ ∈ P 2 (R d ) : supp µ ⊆ S}, we define c(µ (1) , µ (2) , (µ, ν)) = (1) ), and µ |t=T = µ (2) , +∞, otherwise.…”