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

Duprez, Michel; Morancey, Morgan; Rossi, Francesco

doi:10.1109/cdc.2018.8619235

Cited by 6 publications

(6 citation statements)

References 38 publications

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“…As far as the authors know, the aforementioned methods need the whole time interval [0, ∞) for the opinions x i to form a consensus in an asymptotic manner, the object being to stabilize a specific equilibrium of the system. Besides, this is reflected on the infinite dimensional transport model (1.4), whose stabilization properties have been studied, for instance, in [15,26,27].…”

Section: Controllability Properties Of Consensus Modelsmentioning

confidence: 99%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one.We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered.Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.2010 Mathematics Subject Classification. 34D20, 35B36, 65M06, 92D25, 93B05.

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Section: Controllability Properties Of Consensus Modelsmentioning

confidence: 99%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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“…This family of models refers broadly to situations in which a centralised policy-making entity emits a control signal at the macroscopic level, in order to stir an underlying microscopic multi-agent system towards a desired goal (see the introduction of [20] for more details on this general scheme). While a few results have been dealing in this context with controllability issues [49,50] as well as the explicit synthesis of control laws for consensus and alignment models [11,29,30,75], the core of the existing contributions on this topic pertains to mean-field optimal control. In this setting, a first series of articles have aimed at studying rigorously the mean-field limit of solutions of optimal control problems formulated on discrete particle-like systems [38,53,54,55].…”

Section: Introductionmentioning

confidence: 99%

Semiconcavity and Sensitivity Analysis in Mean-Field Optimal Control and Applications

Bonnet,

Frankowska

2021

Preprint

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In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.

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“…Berman et al (2009); Elamvazhuthi et al (2018); Elamvazhuthi and Berman (2019)). A few articles have been dealing with controllability results (see Duprez et al (2019Duprez et al ( , 2020) or explicit syntheses of control laws (e.g. Caponigro et al (2015); Piccoli et al (2015)).…”

Section: Introductionmentioning

confidence: 99%