A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators, policy iteration based on min-computations amounts to consider the characterization of reachable states of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early papers rely on LP to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariant, relying on Semi-Definite Programming (SDP) solvers to compute the next policy, and LP solvers to solve the selected policy. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) optimizations. Our approach enables the precise analysis of switched systems with polynomial assigns and guards. The analysis presented has been implemented in Matlab and applied on existing programs, improving both the set of systems analyzable and the precision of analyzed ones.