This paper investigates the controllability properties of a Shear beam model
without rotary inertia, extending previous research on its stability
characteristics. While recent studies [28] have shown that this system lacks
natural exponential stability when damping is applied only to the angle
rotation, we demonstrate that the introduction of piezoelectric actuators
significantly enhances the system`s controllability. We consider the Shear
beam model governed by coupled partial differential equations for transverse
displacement and rotation angle, subject to hinged boundary conditions. Our
main result establishes the exact L$^{2}$-controllability of this system for
any positive time $T$, using piezoelectric actuators placed at specific
locations $\left( \xi _{1},\mu _{1}\ \right) ,\ \left( \xi _{2},\mu _{2}\
\right) $ along the beam. The proof utilizes the Hilbert Uniqueness Method
(HUM) and a generalization of Ingham`s inequality. We derive explicit
conditions on the actuator placements and system parameters that ensure
controllability. Furthermore, we provide a quantitative lower bound for the
control energy required to steer the system between arbitrary initial and
final states in the L$^{2}$ space. This controllability result has
significant implications for the system`s behavior, effectively overcoming
the limitations of non-exponential stability reported in previous work [28].
Our findings open new avenues for designing optimal control strategies and
feedback stabilization schemes for Shear beam models, with potential
applications in vibration suppression and precise motion control in various
engineering contexts. To validate the practical aspects of our theoretical
findings, we perform comprehensive numerical simulations. These simulations
demonstrate the effectiveness of the proposed control strategy under various
initial conditions and actuator configurations, confirming the robustness
and applicability of our results in realistic scenarios.