On Cauchy problem and stability of inversion-free feedforward control of piecewise monotonic Krasnosel’skii–Pokrovskii hysteresis

We consider the non-homogeneous first-order differential equation with hysteresis described by the Krasnoselśkii-Pokrovskii (KP) rate-independent hysteresis operator. Existence and uniqueness of solutions as well as the boundedness of solutions in response to a bounded input are proved. The global stability of the non-homogeneous equation is also investigated. Periodic solutions and their stability are also studied. The differential equation under analysis constitutes the so-called inversion-free feedforward control, which was proposed for mitigating arbitrary rate-independent hysteresis effects in the actuated systems. The experimentally identified non-smooth and non-strictly monotonic hysteresis of a magnetic shape memory alloy (MSMA) actuator serves as a motivating case study. The performed analysis is settled in a series of theorems which are then illustrated by numerical examples.