Dynamics and control for multi-agent networked systems: A finite-difference approach
Entity
UAM. Departamento de MatemáticasPublisher
World Scientific Publishing Co. Pte LtdDate
2019-04-04Citation
10.1142/S0218202519400050
Mathematical Models and Methods in Applied Sciences 29.4 (2019): 755-790
ISSN
0218-2025 (print); 1793-6314 (online)DOI
10.1142/S0218202519400050Editor's Version
https://doi.org/10.1142/S0218202519400050Subjects
Collective dynamics; Semi-discretization of PDEs; Nonlocal diffusion equations; Controllability; Consensus; MatemáticasNote
Electronic version of an article published as Mathematical Models and Methods in Applied Sciences 29.4 (2019): 755-790 https://doi.org/10.1142/S0218202519400050 © 2019 World Scientific Publishing Company https://www.worldscientific.com/worldscinet/m3asRights
© 2019 World Scientific Publishing CompanyAbstract
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 nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal 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
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Google Scholar:Biccari, Umberto
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Ko, Dongnam
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Zuazua Iriondo, Enrique
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