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The development of control theory from the centralized paradigm to the decentralized one, though explored for many decades, has now become critically important. Indeed, an increasing number of problems of practical and theoretical nature in engineering, biology and communications are characterized by an underlying network topology describing which interactions within a system or between agents are allowed. Such problems include information transmission and gossiping, distributed computation, the study of metabolic networks, etc. We will in this talk focus on the oft-encountered and fundamental problem of stability of a system. The question we seek to answer is, broadly speaking, the following: "can a given network topology sustain stable dynamics?" We will look at the case of linear dynamics and provide a set of sufficient and a set of necessary conditions for the existence of a stable system over a given topology. Furthermore, we will discuss some structural properties of the set of topologies which admit stable dynamics. The conditions we exhibit show how concepts from graph theory can be used to answer questions relating to the dynamics of a system.
M.-A, Belabbas is an Assistant Professor in the ECE department and in the Coordinated Science Laboratory at the University of Illinois At Urbana-Champaign. He received his PhD in Applied Mathematics from Harvard University and was a postdoctoral fellow and lecturer at Harvard and Yale prior to moving to Champaign. He was a visiting Fellow of the Newton Institute in Cambridge, UK in the Spring of 2008. His research interests are in control theory and its applications, and in applied statistics.