Approximate verification of a class of adaptive control systems

We present a procedure for the approximate verification of a class of adaptive control systems. In the systems considered, the continuum state evolves according to a differential equation that is jointly polynomial in states and exogenous inputs. Our computational procedure for approximate verification consists of outer approximating reach sets and checking behavioral inclusion. The main tool for computing outer approximations is a result due to Handelman on the representation of polynomials that are positive in a polytope. It is a special case of Schmudgen's theorem, which is a central result in sum of squares (SOS) programming. Using the result, a linear program for outer approximation is derived. An extension to adaptive systems with state-dependent switching is described. Some difficulties with our approach are also discussed.