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Amir Ali Ahmadi(a_a_amit.edu) Abstract: We introduce the framework of pathcomplete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called pathcomplete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimumofquadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of pathcomplete graphs including a family of dual graphs and all pathcomplete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of pathcomplete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimumofquadratics Lyapunov functions. Keywords: joint spectral radius, stability of switched systems, linear differential inclusions, finite automata, Lyapunov methods, semidefinite programming Category 1: Applications  Science and Engineering (Control Applications ) Category 2: Combinatorial Optimization (Approximation Algorithms ) Category 3: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Citation: Submitted for publication. Download: [PDF] Entry Submitted: 11/22/2011 Modify/Update this entry  
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