In the first part of this talk I will briefly discuss the multi-scale modeling of self-organized systems, with regard to application in social phenomena like swarming, opinion and crowds dynamics. In this context, I will introduce a class of optimal control problems, in order to promote the emergence of a desired state, through centralized or sparse strategies. Due to the presence of a large number of interacting agents, the numerical treatment of this type of problems is in general affected by the curse of high-dimensionality. Therefore I will propose a stochastic method based on the binary approximation of the microscopic dynamics, which will reduce considerably the computational cost. The consistency of this numerical approach is provided in the context of standard kinetic theory, and it is based on the grazing collision limit of the Boltzmann-Povzner equation. Several simulations and numerical examples will show the efficiency of the method for the constrained dynamics.
Keywords: Multi-agent systems, optimal control, Boltzmann equation, Vlasov dynamics, Direct Simulation Monte-Carlo methods.
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