New generalizations of the Petersen lemma (Q463392)

Physical control systems inevitably contain uncertainties and one of the principal models is that of matrix uncertainty. The Petersen lemma is one of the basic tools for handling this model. Let \(G\) be a real symmetric matrix and consider its perturbation \(G+M\Delta N+NT\Delta TMT\), where \(\Delta\) is the perturbing matrix (neither symmetric nor even square) and \(M, N\) are constant nonzero matrices (of corresponding sizes). The Petersen lemma gives a necessary and sufficient condition for the sign-definiteness of the above perturbation under all spectral-norm bounded perturbing matrices \(\Delta\), which reduces the problem to the solvability of a certain matrix inequality involving a scalar variable. An earlier paper by the author and \textit{P. S. Shcherbakov} [Autom. Remote Control 69, No. 11, 1932--1945 (2008); translation from Avtom. Telemekh. 2008, No. 11, 125--139 (2008; Zbl 1163.93021)] dealt with generalizations of Petersen's lemma to cases with more than one matrix uncertainty. In this paper, the author considers the question of other conditions on \(\Delta\) apart from 2-norm boundedness, where the Petersen lemma remains valid.