Oil drilling inspired compensation of wave actuator dynamics for nonlinear systems

The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. We establish global asymptotic stability of the closed-loop system with the aid of a Lyapunov functional that we construct by introducing an infinite-dimensional backstepping transformation of the actuator state.