Towards a Proof of the Kalman Conjecture for the Second Order Systems with Time-Delay

We believe that second-order systems with time delay satisfy the Kalman conjecture. Recently, we have submitted a paper constructing Zames-Falb multipliers for several cases [1], but a formal proof is still to be developed. This abstract summarises recent development towards a formal proof of the above statement. The construction of the multiplier is irrational, and the proof cannot be developed as in the non-delay case as it uses the pole location of the closed-loop system at the Nyquist gain. Hence a proof would be significantly different from the current proofs provided of the Kalman conjecture for different cases [2]–[4].