Positive real lemma: Necessary and sufficient conditions for the existence of solutions under virtually no assumptions

In this note, the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system-theoretic assumptions. More precisely, necessary and sufficient conditions are established under the very weak assumption of sign-controllability. It is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem. This reduction, beside being interesting per se, permits us to weaken the assumption of sign-controllability. Finally, the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis.