KINETIC APPROXIMATION, STABILITY AND CONTROL OF COLLETIVE BEHAVIOR IN SELF-ORGANIZED SYSTEMS

Abstract

The aim of this thesis mainly focus on the study of self-organized systems, from different level of descriptions. We develop efficient numerical methods based on Direct Simulation Monte Carlo techniques, to solve the kinetic approximation of these systems, with a considerable save in the computational cost. We study the stability and instability of flock ring and mill ring solutions of second order swarming models, determining the stable regions of parameters and obtaining a spectral equivalence between first order and second order model in the case of flock solutions. In the second part we embed classical swarming models with control dynamics. We first present a general framework for swarming model interacting with few individuals, seen as external point source forces, giving a microscopic, a mesoscopic and a macroscopic description. Later we focus on optimal control problems for self-organized systems and inspired by model predictive control strategy, we obtain a kinetic description of the initial optimal control problem. In the end we report an asymptotic preserving scheme or optimal control problems of boundary problems governed by the hyperbolic relaxed systems.