In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discrete-time model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.

this paper is to give a brief tutorial review of four different classes of hybrid systems of this type - each consists of a continuous-time process to be controlled, a parameterized family of candidate controllers, and an event driven switching logic. Three of the logics, called prerouted switching, hysteresis switching and dwelltime switching respectively, are simple strategies capable of determining in real time which candidate controller should be put in feedback with a process in order to achieve desired closed-loop performance. The fourth, called cyclic switching, has been devised to solve the long-standing stabilizability problem which arises in the synthesis of identifier-based adaptive controllers because of the existence of points in parameter space where the estimated model upon which certainty equivalence synthesis is based, loses stabilizability

Asymptotic Properties of Nonlinear Feedback Control Systems A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts by Daniel Liberzon We study asymptotic behaviour of nonlinear feedback control systems, both deterministic and stochastic. Of particular interest is the case of quantized feedback, i.e., when the nonlinearity takes the form of a specific piecewise constant function. In the context of deterministic linear control systems with quantized measurements, we show how quantized feedback can be used to asymptotically stabilize the system (Chapter II). For systems perturbed by white noise, we address the question of existence of steady-state probability distributions. In the linear case, the solution to the v Fokker-Planck equation which describes the evolution of the probability density is well known. In particular, one has an expression for the steady-state probability density, which is an eigenfunction...

This thesis focuses on the analysis of emergent behaviors of a multi-agent system. The ultimate aim is to synthesize complex group behaviors from simple social inter-actions among individuals based on simple strategies. One of the many challenges in the study of emergent behaviors of multi-agent systems is characterizing the local properties of each agent that will lead to global emergent behaviors. Two types of emergent behaviors are described, namely, the Multi-agent Flocking Problem and the Multi-agent Rendezvous Problem. In both problems, provably correct distributed local rules are proposed. In the Flocking problem, a local heading average rule was studied and convergence of the group heading to a common unspecified heading was proved using results in convergence of matrix products in Markov Chains. Replacing the word agent by a more general notion of identity naturally leads the Flocking problem to a class of network consensus problems. In the Rendezvous problem, convergence of the group positions to a common unspecified location was established using combinations of convex analysis and standard Lyapunov function approach. An asynchronous version of the Rendezvous problem was also solved by modelling the process using a suitably defined hybrid model and a procedure called Analytic Synchronization. Recent applications of the Controlled Mobility concept to wireless network in both synchronous and asynchronous settings to optimize energy consump-tion is briefly discussed at the end.

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