On boundary controllability of a vibrating plate

A vibrating plate is here taken to satisfy the model equation:u_tt + ²u = 0 (where²u:= (u); = Laplacian) with boundary conditions of the form:u_v = 0 and(u)_v = = control. Thus, the state is the pair [u, u_t] and controllability means existence of on := (0,T)× transfering any[u, u_t]₀ to any[u, u_t]_T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T^–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u_tt = u) with log = O(T^–1) asT0. Some related results on minimum time control are also included.This research was partially supported under the grant AFOSR-82-0271.