Dual Lyapunov stability analysis in behavioral approach

This paper considers a Lyapunov stability analysis for continuous-time systems described by high order difference-algebraic equation from the viewpoint of the semidefinite programming (SDP) duality. In the behavioral system theory, a Lyapunov function is described by a quadratic differential form (QDF) and equivalently characterized by a two-variable polynomial matrix. We first develop the SDP duality to the non-negativity and positivity of two-variable polynomial matrices. Using the duality, we derive an alternative stability condition in terms of the two-variable polynomial matrix equation and QDFs as a main result.