Mean-Field Pontryagin Maximum Principle

Bongini, Mattia; Fornasier, Massimo; Rossi, Francesco; Solombrino, Francesco

doi:10.1007/s10957-017-1149-5

Cited by 67 publications

(76 citation statements)

References 45 publications

(62 reference statements)

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“…It is therefore natural to seek controls achieving the same goal with less active components. This is the case for instance when we want only one leader to act on a whole crowd (such as a dog with a flock of sheep), or more generally when feasible control strategies are required to focus on a small number of agents at each time (see [1,2,6,7,20,33]). …”

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confidence: 99%

Sparse Jurdjevic–Quinn stabilization of dissipative systems

et al. 2017

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For control-affine systems with a proper Lyapunov function, the classical Jurdjevic-Quinn procedure (see [22]) gives a well-known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated, i.e. nonzero, at the same time.In this paper we give sufficient conditions under which this stabilization can be achieved by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. We thus obtain a sparse version of the classical Jurdjevic-Quinn theorem.We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying periodic feedback, a sampled feedback, and a hybrid hysteresis.We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models.

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confidence: 99%

Sparse Jurdjevic–Quinn stabilization of dissipative systems

et al. 2017

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“…Even though the evaders are gathered initially, the pressure from the drivers may violate the flocking of the evaders. In Figure 17, the trajectories of the system and other indicators are shown in the simulation with 16 evaders, leading to the target (4,4). We can observe that the diameter of evaders' position increases in the pursuing dynamics.…”

Section: 2mentioning

confidence: 94%

“…In [17], feedback formation is suggested to collect sheep in a small area. In [4], the well-posedness for optimal control problems is established on transport equations of the herd, coupled with ordinary differential equations (ODEs) of the dogs.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior and control of a “guidance by repulsion” model

Zuazua

2020

Math. Models Methods Appl. Sci.

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We model and analyze a herding problem, where the drivers try to steer the evaders' trajectories while the evaders always move away from the drivers. This problem is motivated by the guidance-by-repulsion model [10], where the authors answer how to control the evaders' positions and what is the optimal maneuver of the drivers. First, we obtain the well-posedness and the long-time behavior of the one-driver and one-evader model, assuming of the same friction coefficients. In this case, the exact controllability is proved in a long enough time horizon. We extend the model to the multi-driver and multi-evader case, and develop numerical simulations to systematically explore the nature of controlled dynamics in various scenarios. The optimal strategies turn out to share a common pattern to the one-driver and one-evader case: the drivers rapidly occupy the position behind the target, and want to pursuit evaders in a straight line for most of the time. Inspired by this, we build a feedback strategy which stabilizes the direction of evaders.Date: November 5, 2019.

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“…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,).…”

Section: Introductionmentioning

confidence: 99%

Leader formation with mean-field birth and death models

Albi

Bongini²,

Rossi

et al. 2019

Math. Models Methods Appl. Sci.

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We provide a mean-field description for a leader-follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions for the leader-follower dynamics, under suitable assumptions. We then establish, for an appropriate choice of the initial datum, the equivalence of the system with a PDE-ODE system, that consists of a continuity equation over the state space and an ODE for the transition from leader to follower or vice versa. We further introduce a stochastic process approximating the PDE, together with a jump process that models the switch between the two populations. Using a propagation of chaos argument, we show that the particle system generated by these two processes converges in probability to a solution of the PDE-ODE system. Finally, several numerical simulations of social interactions dynamics modeled by our system are discussed.

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Mean-Field Pontryagin Maximum Principle

Cited by 67 publications

References 45 publications

Sparse Jurdjevic–Quinn stabilization of dissipative systems

Sparse Jurdjevic–Quinn stabilization of dissipative systems

Asymptotic behavior and control of a “guidance by repulsion” model

Leader formation with mean-field birth and death models

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