L_{2}-stability of linear time-varying systems--Conditions involving noncausal multipliers

Abstract
New criteria in the multiplier form are presented for the input-output stability in the L 2 -space of a linear system with a time-varying element k(t) in a feedback loop. These are sufficient conditions for the system stability and involve conditions on the shifted imaginary-axis behavior of the multipliers. The criteria permit the use of noncausal multipliers, and it is shown that this necessitates dk/dt to be bounded from above as well as from below. The method of derivation draws on the theory of positivity of compositions of operators and time-varying gains, and the results are shown to be more general than the existing criteria.