Stabilizing Nonlinear ODEs With Diffusive Actuator Dynamics

This paper presents a design of stabilizing controllers for a cascaded system consisting of a boundary actuated parabolic PDE and nonlinear dynamics at the unactuated boundary. Although the considered PDE is linear, the nonlinearity of the ODE constitutes a significant challenge. In order to solve this problem, it is shown that the classical backstepping transformation of Volterra type directly results from the solution of a Cauchy problem. This new perspective enables the derivation of a controller for the nonlinear setup, where a Volterra integral representation does not exist. Specifically, the solution of an appropriate linear Cauchy problem yields a novel state transformation facilitating the design of a stabilizing state feedback. This control law is shown to ensure asymptotic closed-loop stability of the origin. An efficient implementation of the controller is proposed and demonstrated for an example.