Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment

AbstractWe consider the Euler-Bernoulli problem (1.1) in the solution w(t, ·) with boundary controls g1 and g2 acting in the Dirichlet traces for w and Δw, respectively. We show two main exact controllability results both in the spaces of maximal regularity of the solution at t = T (see I. Lasiecka and R. Triggiani, Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Unione Mat. Ital. (7), 3-B (1989).) and both without geometrical conditions on the open bounded domain Ω (except for smoothness of its boundary Γ): one with g1 ϵ L2(0, T; L2(Γ)) and g2 ϵ [H1(0, T; L2(Γ))]′ for any T > 0 arbitrarily short; and one with g1 ϵ H01(0, T; L2(Γ)) and g2 ϵ L2(0, T; L2(Γ)) for all T > 0 sufficiently large. An interpolation result between these two cases is also presented. A direct approach is given based on two main steps. First, by means of an operator model (I. Lasiecka and R. Triggiani, Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Unione Mat. Ital. (7), 3-B (1989).) for problem (1.1) and a functional analytic approach, the question of exact controllability is shown to be equivalent to an a-priori inequality for the corresponding homogeneous problem. Next, this a-priori inequality is proved to hold true by means of multiplier techniques. These are inspired by recent progress on the maximal regularity, exact controllability and uniform stabilization questions for second order hyperbolic equations