Discretized Jensen's inequality: an alternative vision of the reciprocally convex combination lemma

The analysis of time-varying delay systems has attracted many researchers over the last two decades. One of the major contribution within this field is the recent reciprocally convex combination lemma. The relevance of this lemma arises from the derivation of stability conditions using Jensen's inequality. The main interest of this lemma is the reduction of the number of decision variables while keeping the same level of conservatism. In this paper, we provide an alternative vision of this inequality through a new proof issued from the recent development on integral inequalities. The benefit of this proof relies on the derivation of an exact expression of the conservatism. A discussion is finally proposed at the end of the paper to point out the possible extensions of this approach.