Adaptive Stabilization for ODE–PDE–ODE Cascade Systems With Parameter Uncertainty

In this article, we study the adaptive stability for parabolic partial differential equation (PDE)-ordinary differential equation (ODE) cascade systems with actuator dynamics, where the actuator dynamics are nonlinear subject to unknown parameters. Compared with a class of PDE–ODE coupled systems that the control input only acts on the PDE boundary and the linear sandwiched system without uncertainty, the structure of such systems is more complex. First of all, infinite-dimensional backstepping transformation is adopted. The original PDE-ODE cascade system is changed to a new system that is easier to design. On this basis, finite-dimensional backstepping transformation and adaptive compensation technology are combined to develop a state-feedback controller. Then, the boundedness of all the signals in the closed-loop system is proved by the Lyapunov functional analysis. Furthermore, the control law and the original system states eventually converge to zero. Finally, different simulation data are presented to illustrate the validity of the theoretical results.