Alex Olshevsky

Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R^{n x n} are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k. We show that if $k= \lceil n^d \rceil$ for any fixed real d>0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P is not NP, a widely believed conjecture in computer science. On the other hand, when k is a constant independent of n, then it is known that the problem may be solved in polynomial time in n. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than n^d matrices A_i in R^{n x n} satisfying 0 >= A_i + A_i^{T} is NP-hard.

We propose three new algorithms for the distributed averaging and consensus problems: two for the fixed-graph case, and one for the dynamic-topology case. The convergence rates of our fixed-graph algorithms compare favorably with other known methods, while our algorithm for the dynamic-topology case is the first to be accompanied by a polynomial-time bound on the convergence time

We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.