Abstract
In this paper we investigate a time-optimal control problem in the space of positive and finite Borel measures on \begin{document}$\mathbb R^d$\end{document} , 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.
Highlights
To include uncertainty features in control problems, researchers used a set of different approaches, such as deterministic [16], random [23] and stochastic [4, 25], and applied to different domains as finance [18] and quantum control [6].In particular, in stochastic approaches the state is represented by a random variable or, alternatively, a probability distribution
The evolution is given by an equation involving Brownian motion and solution is interpreted in the sense of the Ito or Stratonovich integral [17]
The paper is structured as follows: Section 2 recalls some preliminary results and notation; in Section 3 we define the clock-admissible trajectories involved in our study and prove some approximation results on the mass-preserving trajectories on which our objects are built; in Section 4 we state the time-minimization problem, in Theorem 4.4 we prove the existence of an optimal clock-trajectory constructing it by approximation techniques, and we conclude by stating a dynamic programming principle
Summary
To include uncertainty features in control problems, researchers used a set of different approaches, such as deterministic [16], random [23] and stochastic [4, 25], and applied to different domains as finance [18] and quantum control [6]. Given a target set S ⊆ Rd closed and nonempty, and recalling the definition of classical minimum time function T : Rd → [0, +∞], an admissible trajectory γis called optimal for x ∈ Rd if γ(0) = x, γ(T (x)) ∈ S. For any n ∈ N we have that μn = {μnt }t∈[0,Tn] is an admissible mass-preserving trajectory defined on [0, Tn], starting from μ, driven by νn := {νtn}t∈[0,Tn], and represented by ηn; 3. In Theorem 4.4, we will see how to construct an optimal-clock trajectory by approximation techniques, in particular by using Lusin’s theorem and Corollary 2 This result will allow us to express τ (μ) as an average of the classical minimum-time function T (·). Consider a continuous selection vi+1 of F and a solution {μit+1}t∈[0,ε] of
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