We consider the mixed problem for the “Kirchhoff plate equation” on an open bounded domain \(\Omega \) in \(\mathbb {R}^n, n = 1, 2, 3,, \ldots \) with sufficiently smooth boundary \(\Gamma = \partial (\Omega )\) . Under both Dirichlet and Neumann homogeneous Boundary Conditions, the dynamical system defines a strongly continuous group of unitary operators on an appropriate function space. We then introduce a suitably devised Neumann boundary control in feedback form, as to force the new dynamic (to be well-posed and) to asymptotically decay in an optimal function space (the same space of optimal regularity and exact controllability under open-loop control, \(L^2\) -in time and space.) We obtain the following results: (1) uniform stabilization for \(n=1\) ; (2) polynomial/rational stability for \(n = 2,3,4, \ldots \) ; (3) and, independently, strong stabilization for any dimension n. In the present paper, we employ a frequency domain approach, based on technical PDE-estimates.
