  By computing Lyapunov functions of a certain, convenient structure,
Lyapunov-based methods guarantee stability properties of the system or, when
performing synthesis, of the relevant closed-loop or error dynamics. In doing
so, they provide conclusive affirmative answers to many analysis and design
questions in systems and control. When these methods fail to produce a feasible
solution, however, they often remain inconclusive due to (a) the method being
conservative or (b) the fact that there may be multiple causes for
infeasibility, such as ill-conditioning, solver tolerances or true
infeasibility. To overcome this, we develop LMI-based theorems of alternatives
based upon which we can guarantee, by computing a so-called certificate of
nonexistence, that no poly-quadratic Lyapunov function exists for a given
linear parameter-varying system. We extend these ideas to also certify the
nonexistence of controllers and observers for which the corresponding
closed-loop/error dynamics admit a poly-quadratic Lyapunov function. Finally,
we illustrate our results in some numerical case studies.
