Contrôle interne exact des vibrations d’une plaque rectangulaire. (Internal exact control for the vibrations of a rectangular plate). (French) Zbl 0718.49026

Jaffard, S.

Port. Math. 47, No. 4, 423-429 (1990).

Consider the equation \((\partial^ 2u/\partial t^ 2)+\Delta^ 2u=h(x,t)\) of the control of a vibrating plate of rectangular shape with Dirichlet boundary conditions on u and \(\Delta\) u. It is proved in this paper that any initial data \((u,\partial u/\partial t)_{t=0}\) belonging to \((H^ 2\cap H^ 1_ 0)\times L^ 2\) can be driven to rest in an arbitrarily small time by a control h(x,t) in \(L^ 2\) supported by an arbitrarily small open subset of the rectangle. This result is derived from classical estimates on lacunary Fourier series.

Reviewer: S.Jaffard

MSC:

49N10	Linear-quadratic optimal control problems
35K25	Higher-order parabolic equations
93B05	Controllability
74H45	Vibrations in dynamical problems in solid mechanics

Keywords:

vibrating plate; Dirichlet boundary conditions; lacunary Fourier series