This talk is devoted to the stabilization problem for flexible beam
oscillations. The considered mechanical system consists of a simply supported
flexible beam of the length l with collocated piezoelectric actuators and
sensors and a controlled spring-mass attached at the point l0 of the beam.
The beam's oscillations are modeled as the Euler-Bernoulli equation with
interface and boundary conditions. Mathematical model is derived using
Hamilton's principle. The equation of motion is presented in the form of an
abstract differential equation with control
$$\xi˙ = A\xi + Bu$$
in the Hilbert space $X = H^2(0, l)\times L^2(0, l)\times R^2$ with fourth-order differential
operator $A : D(A) \to X$. It is proved that the operator $A$ is the infinitesimal
generator of a $C_0$-semigroup in $X$. A feedback control is obtained in such a
way that the total energy is non-increasing on the closed loop system
trajectories. It is shown, based on Lyapunov's theorem, that the obtained
feedback stabilizes the coupled distributed and lumped parameter system. The
resolvent of the infinitesimal generator is obtained and is proved to be
compact. So, the trajectories of the closed loop system form a precompact set in X.
The main result is the proof that the infinite-dimensional closed loop system
has asymptotically stable trivial equilibrium under some natural assumptions on the
mechanical structure. This result is obtained with the help of LaSalle's
invariance principle. The spectral problem is studied for estimating the
distribution of the beam's
eigenfrequencies. The observation problem for linear system is investigated
for nite-dimensional projections of the original dynamical system. A
Luenberger-type observer is designed
for the system derived by Galerkin's approximations. The observer gain
parameters are defined in such a way that the observer properly estimates the
complete state vector,
i.e. the error dynamics has asymptotically stable trivial equilibrium.
Results on the eigenfrequencies distribution and observer convergence are
illustrated by numeric simulations.