Observer-based stabilization of an Orr-Sommerfeld equation cascaded by both the Squire equation and ODE with boundary control

Abstract

In this paper, we study the boundary output feedback stabilization for an Orr-Sommerfeld equation cascaded by both the Squire equation and ordinary differential equations. The observer is proposed and the gain functions are found by the backstepping method. The observer-based controller is designed by using the certainty-equivalent principle. The existence and exponential stability of the closed-loop system are proved by calculating the system operator to generate a C0-semigroup and its eigenfunctions to form the Riesz basis approach.