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By computing Lyapunov functions of a certain, convenient structure,Lyapunov-based methods guarantee stability properties of the system or, whenperforming synthesis, of the relevant closed-loop or error dynamics. In doingso, they provide conclusive affirmative answers to many analysis and designquestions in systems and control. When these methods fail to produce a feasiblesolution, however, they often remain inconclusive due to (a) the method beingconservative or (b) the fact that there may be multiple causes forinfeasibility, such as ill-conditioning, solver tolerances or trueinfeasibility. To overcome this, we develop LMI-based theorems of alternativesbased upon which we can guarantee, by computing a so-called certificate ofnonexistence, that no poly-quadratic Lyapunov function exists for a givenlinear parameter-varying system. We extend these ideas to also certify thenonexistence of controllers and observers for which the correspondingclosed-loop/error dynamics admit a poly-quadratic Lyapunov function. Finally,we illustrate our results in some numerical case studies.

Certificates of Nonexistence for Lyapunov-Based Stability, Stabilizability and Detectability of LPV Systems