Asymptotic stability of linear systems with restricted connectivity

We investigate the conditions under which a given network topology can support an asymptotically stable equilibrium in linear systems, in both discrete and continuous time. This question arises naturally in the study of decentralized networked systems, where stability properties are constrained by the underlying connectivity structure. The study of such systems was initiated by M.-A. Belabbas approximately a decade ago, and he had obtained some conditions for stability. I will first present the key results of his theory, and then present our recent work on the switching (nonautonomous) case, where system dynamics can be time-varying but with the fixed network topology. Finally, I will outline preliminary results on generalizing these ideas to certain classes of nonlinear systems. This is joint work with M.-A. Belabbas and D. Liberzon (ECE), and R. Bivziuk (Mathematics).