Input-dependent incremental stability criterion for piece-wise linear analogs of van der Pol systems

The paper proposes and advocates a novel technique of Lyapunov analysis that is based on the concept of averaging functions and Steklov´s averaging method. This approach relaxes quadratic criteria for incremental stability of nonlinear systems. The stability criteria rest on time-dependent quadratic or non-quadratic Lyapunov functions and offer better account for the role of the external excitation by providing input-dependent conditions. In particular, the proposed method works even if the system fails to satisfy the incremental version of the circle criterion and, for example, proves frequency entrainment for a piece-wise linear analog of the van der Pol oscillator whenever the input has sufficiently large amplitude.