Boundary stabilization of two-dimensional Petrovsky equation: Vibrating plate (Q810734)

The author considers the plate equation \(w_{tt}+\Delta^ 2w=0\) on the unit square \([0,1]^ 2\) with one side clamped, the adjacent sides supported and the opposite side dynamically controlled by \(w_{tt}- \beta_ 1D^ 3_ xw+(\sigma -2)D_ xD^ 2_ yw=f_ 1,D_ xw_{tt}+\beta_ 2D^ 2_ xw-\sigma D^ 2_ yw=f_ 2\) on \(\{x=1\}\). He shows that one can write this problem, letting \(v=(w,w|_{x=1}\), \(D_ xw|_{x=1})\) and \(u=(v,v_ t)\), \(f=(f_ 1,f_ 2)\), in the abstract form \(u_ t=Gu+Kf\), \(u(0)=u_ 0\), where G is the infinitesimal generator of a unitary \(C_ 0\)-group. The main result is, that one gets \(S(t)u_ 0\to 0\), if S(t) denotes the semigroup generated by \(G-KK^*\). Hence taking the linear feedback control \(f=-K^*u=-(w_ t,D_ xw_ t)|_{x=1}\) the system is strongly stabilized.