Improvement of the ARE strict bounded real lemma

The existing ARE version of the strict bounded real lemma, which establishes equivalence between a strict H
_infin
norm bound condition and the existence of a positive semidefinite stabilizing solution of an ARE, comes with an a priori stability hypothesis which makes its use cumbersome. This paper proposes an improved version of the lemma where the stability condition is not as a hypothesis but instead goes with the norm bound condition. The improved version offers nothing less than the existing one while providing additionally the fact that the stability is already ensured by the fulfilment of the ARE condition. Similar improvements are also done to the positive real lemma and the L
_infin
version of the bounded/positive real lemma.