Sliding Mode Control of the Orr--Sommerfeld Equation Cascaded by Both the Squire Equation and ODE in the Presence of Boundary Disturbances

The sliding mode control (SMC) approach is proposed in this paper for the Orr--Sommerfeld equation cascaded by both the Squire equation and ODE, in which Orr--Sommerfeld and Squire equations are subject to boundary controls matched disturbances, respectively. A modified backstepping transformation with matrix kernel is integrated with SMC to reject the bounded disturbances. The well-posedness of the solution for the closed-loop system is obtained based on the Riesz basis method and the finite time “reaching condition” is verified. Finally, the numerical simulations are presented to illustrate the effectiveness of the control approach.