Stability and stabilization of the generalized Benney-Lin equation with an antidamping term using bounded-input feedback control

The control problem of the generalized Benney-Lin equation with an antidamping term is considered. This is done through a bounded input feedback control design. First, we show that the system under the effect of an antidamping function is well-posed. Then, we demonstrate that the stability of the system depends on two parameters in the equation; the positive parameter ν and the bounded antidamping function \(d(x)\) . The stability analysis is conducted using Lyapunov indirect method. That is, local stability results are drawn based on the stability of the linearized generalized Benney-Lin equation around the zero equilibrium solution. Then, a bounded input feedback control design is presented where the controller is originally designed to the linearized system and then implemented to the nonlinear equation not only to achieve local stability result of the solution, but also to speed up the convergence rate when the system is already stable. Finally, some numerical examples to illustrate the approach are presented.