Summary: Given \(\Omega\), an open bounded domain in \(\mathbb{R}^n\) with sufficiently smooth boundary \(\Gamma\), we consider the nonhomogeneous Euler-Bernoulli equation in \(w(t, x)\): (1) \(w_{tt}+ \Delta^2 w= 0\) in \(Q= (0, \infty)\times \Omega\), \(w(0, \cdot)= w_0\); \(w_t(0, \cdot)= w_1\) in \(\Omega\), \(w|_\sigma= g_1\in L^2(\Sigma)= L^2((0, \infty); L^2(\Gamma))\) on \(\Sigma= (0, \infty)\times \Gamma\), \(\partial w/\partial v|_\Sigma= g_2= 0\) on \(\Sigma\).
We express the nonzero control function \(g_1\) as a suitable linear feedback applied to the velocity \(w_t\), i.e. \(w|_\Sigma= \mathbb{F} w_t\), such that \(\mathbb{F} w_t\in L^2((0, \infty); L^2(\Gamma))\), and the corresponding closed loop system obtained by using such a feedback in (1) generates a (feedback) \(C_0\)-semigroup \(e^{\mathbb{A} t}\) on \(Z\) which decays strongly to zero: \(|e^{\mathbb{A} t} z|_Z\to 0\) as \(t\to \infty\), for all \(z\in Z\), where \(Z= [D(A^{1/4})]'\times [D(A^{3/4})]'\).
Having identified a candidate \(\mathbb{F} w_t= -(\partial/\partial v)[\Delta (A^{3/2} w_t)]\), where \(A\) is the operator defined by \(Af= \Delta^2 f\); \(D(A)= \{f\in L^2(\Omega): \Delta_2 f\in L^2(\Omega)\), \(f|_\Gamma= \partial f/\partial v|_\Gamma= 0\}\), we prove strong stabilization. Specifically, solutions of (1) go to zero in the strong topology of \(Z\): \(\lim_{t\to \infty}|[w(t), w_t(t)]|_Z= 0\), by the use of a Hilbert space decomposition for contractive semigroups.