  In this paper, we propose a distributed computing approach to solving
large-scale robust stability problems on the simplex. Our approach is to
formulate the robust stability problem as an optimization problem with
polynomial variables and polynomial inequality constraints. We use Polya's
theorem to convert the polynomial optimization problem to a set of highly
structured Linear Matrix Inequalities (LMIs). We then use a slight modification
of a common interior-point primal-dual algorithm to solve the structured LMI
constraints. This yields a set of extremely large yet structured computations.
We then map the structure of the computations to a decentralized computing
environment consisting of independent processing nodes with a structured
adjacency matrix. The result is an algorithm which can solve the robust
stability problem with the same per-core complexity as the deterministic
stability problem with a conservatism which is only a function of the number of
processors available. Numerical tests on cluster computers and supercomputers
demonstrate the ability of the algorithm to efficiently utilize hundreds and
potentially thousands of processors and analyze systems with 100+ dimensional
state-space. The proposed algorithms can be extended to perform stability
analysis of nonlinear systems and robust controller synthesis.
