On the Frontier between Kalman Conjecture and Markus-Yamabe Conjecture

The Kalman conjecture and the Markus-Yamabe conjecture are revisited. These conjectures make statements about global stability of certain classes of nonlinear systems. Both conjectures are known to be false in the general case, but in the context of traction control, an interesting special case of the Markus-Yamabe conjecture becomes apparent. In this letter, two methods are proposed to analyze general Lur'e systems with respect to their input and output dimension, resulting in a new open problem in the context of the Markus-Yamabe conjecture. The connection to ridge functions and traction control systems is discussed. It is shown that although both the Kalman conjecture, for single-input, single-output Lur'e systems and the Markus-Yamabe conjecture are true up to a specific system order, the problem is still open if different input and output dimensions of a Lur'e formulation are considered.