Energy Decay Estimate for a Wave-Plate Interface Transmission Problem with Only Two Dynamical Boundary Controls

In Abdallah et al. (Acta Applicandae Mathematicae 189, 2024), Abdallah et al. considered the stabilization of a transmission model consisting of a wave equation and a Kirchhoff plate equation. They accomplished this by employing three dynamical boundary controls while imposing specific geometric conditions. However, the problem of stabilizing the system when it is damped by only two dynamical boundary controls remained open. In this chapter, we estimate the decay rate of the energy under the condition where the dynamical control \(\xi \) is omitted, thereby subjecting the system to the influence of two dynamical controls, namely \(\eta \) and \(\zeta \) . To achieve this, we use a unique continuation theorem to establish the strong stability of the system under geometrical conditions concerning the plate’s domain. However, we establish a polynomial energy decay estimate of type \(1/t\) for smooth initial data. This is accomplished using the frequency domain approach, which in turn imposes certain geometric conditions relating to the wave’s domain.