A new approach to the Lur´e problem for non-autonomous systems with arbitrary delay

The classical Lur´e problem consists in finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector. Most of the results obtained on this problem are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach providing a sufficient stability criterion for systems with a non-autonomous linear block and an arbitrary time-varying delay in the feedback. The result is expressed in the transfer function of the linear block and the sector margins of the nonlinear block. It is shown that stability of a system with a sign-constant transfer function is guaranteed by stability of the system with a limit linear feedback (so that, for such systems, the famous Aizerman conjecture is true).