Stabilization of gravity-capillary water waves with constant vorticity

We establish an exponential stabilization result for two-dimensional periodic gravity-capillary water waves of finite depth with constant vorticity, using an external pressure localized on the free surface. The pressure is prescribed as the product of a nonnegative cut-off function and the time derivative of the surface elevation and serves as a feedback control. To prove the main result, we derive the energy dissipation identity and estimate the time integral of the energy relative to the flat Couette flow, by means of the multiplier method and periodic Pohozaev identities. A remaining lower-order term led by the vorticity is removed by a compactness argument based on unique continuation for the linearized equations. Under a condition relating the cut-off to gravity and surface tension, the relative energy decays exponentially on any interval on which a regular solution remains uniformly small in Sobolev norms. We do not assume the evenness on the solution and smallness on the vorticity.