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. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H
₂
× H
₁
and C
¹
× C
⁰
norms for general nonlinear ODEs, as well as in the H
₁
× L
₂
norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring,” as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.