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Asym­pto­tic Sta­bi­liza­t­i­on of the Fle­xi­ble Beam Oscil­la­ti­ons

Livestream:  https://uni-kassel.zoom.us/j/96217091997?pwd=RVRaVVBRclFJYU9jczNZSWF3SXI2QT09
Meeting ID: 962 1709 1997
Passcode: cauchy

Asymptotic Stabilization of the Flexible Beam Oscillations
M.Sc. Julia Kalosha (NAS Ukraine & MPI Magdeburg)

Abstract:
The presented results are joint work with Prof. Alexander Zuyev, Max Planck Institute for Dynamics of Complex Technical Systems (Magdeburg).

Mechanical structures with flexible beams are widely used in modern engineering, in particular, in the fields of spacecraft manufacturing, large-scale robotics, wind-turbine industry, and offshore drilling technology. Parallel with rapid ongoing technological progress, the line between the control engineering and the mathematical control theory is blurring. Inspired by industrial challenges, the control theory of elastic systems with distributed parameters has been refined over the several last decades.

This talk is devoted to the stabilization problem for a dynamic system which describes the vibrations of an elastic beam with an attached rigid body and distributed control actions. The mathematical model is derived using Hamilton's principle in the form of the Euler-Bernoulli beam equation with hinged boundary conditions and interface condition at the point of attachment of the rigid body.

The equation of motion is presented in the form of an abstract differential equation with control

ξ' = Aξ + By

in the Hilbert space H²₀(0, ℓ) × L²(0, ℓ) × ℝ² with fourth-order differential operator A : D(A) → X.

Unlike the Cauchy problem for finite-dimensional ODE systems with smooth right-hand side, the conditions of well-posedness for infinite-dimensional systems become one of the major tasks. For dealing with this the semigroup representation in Banach or Hilbert spaces is commonly used. It is proved that differential operator generates a C0-semigroup in X .

The aim is to propose the control y , such that the state of rest ξ = 0 is stable in Lyapunov sense or even asymptotically stable. A feedback control is obtained in such a way that the total energy is non-increasing on the closed-loop system trajectories.

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. Results are illustrated by numeric simulations.

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