  This study focuses on the exact observability of a non-classical
Euler-Bernoulli micro-beam equation. This non-classical model was derived based
on the strain gradient elasticity theory, which is intended to explain the
phenomenon of size effect at the micron scale. Spectral properties of the
corresponding state operator are studied; an asymptotic expression for
eigenvalues is calculated, and eigenfunctions are analyzed in order to check
the necessary conditions for the exact observability of the system. By
examining the eigenfunctions, it is shown that among non-collocated boundary
outputs, only measurement of the non-classical moment at the root of the beam
yields an admissible observation operator and also defines an exactly
observable system. An alternative proof based on the multiplier method, which
is commonly employed in the literature on the observability and controllability
of infinite dimensional dynamical systems, is presented to provide a comparison
between the time and frequency domain approaches.
