# DIRECT APPROACH TO EXACT CONTROLLABILITY FOR THE WAVE EQUATION WITH NEUMANN BOUNDARY CONTROL AND TO AN EULER-BERNOULLI EQUATION.

## Abstract

Let OMEGA be an open bounded domain in R**n with sufficiently smooth boundary GAMMA . The authors study the exact controllability property of both (i) the wave equation defined on OMEGA with control action exercised in the Neumann boundary conditions, and (ii) an Euler-Bernoulli equation defined on OMEGA , likewise with boundary controls. The approach is direct (not through uniform stabilization) and is based in its incipient stage on the obvious requirement that the input-solution operator be subjective (onto) between the space of boundary controls and the target state space, both of which are preassigned in advance. Through an operator approach, the equivalent condition that the (Hilbert space) adjoint of the solution operator have a continuous inverse is identified. Such characterization turns out to be given in terms of a suitable inequality involving an appropriate trace of the solution of the corresponding homogeneous problem, which evolves only by virtue of the initial conditions. The crux of the controllability property consists in showing that such inequality does hold true for suitable T greater than 0.

## Publication Title

Proceedings of the IEEE Conference on Decision and Control

## Recommended Citation

Lasiecka, I., & Triggiani, R.
(1987). DIRECT APPROACH TO EXACT CONTROLLABILITY FOR THE WAVE EQUATION WITH NEUMANN BOUNDARY CONTROL AND TO AN EULER-BERNOULLI EQUATION..* Proceedings of the IEEE Conference on Decision and Control*, 529-534.
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