On the exponential decay of the Euler-Bernoulli beam with boundary energy dissipation

We study the asymptotic behavior of the Euler-Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the "exponential decay" of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy.Comment: 13 page

Gevrey's and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions

AbstractThe main aim of the paper is to present a technique that allows to infer wellposedness, trace and Gevrey's regularity of hyperbolic-like PDE's with non-monotone boundary conditions. The lack of monotonicity prevents applicability of the known semigroup methods.In this paper we show how recently developed tools of microlocal analysis [D. Tataru, A priori estimates of Carleman's type in domains with boundary, J. Math. Pure Appl. 73 (1994) 353–387] combined with some spectral theory can be used successfully in order to obtain the needed inequalities. The method will be illustrated on a simple example of beam equation with non-monotone boundary conditions

Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an EulerBernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable

Robust Output Regulation of Euler-Bernoulli Beam Models

In this thesis, we consider control and dynamical behaviour of flexible beam models which have potential applications in robotic arms, satellite panel arrays and wind turbine blades. We study mathematical models that include flexible beams described by Euler-Bernoulli beam equations. These models consist of partial differential equations or combination of partial and ordinary differential equations depending on the loads and supports in the model. Our goal is to influence the models by control inputs such as external applied forces so that measured deflection profiles of the beams in the models behave as desired. We propose dynamic controllers for the output regulation, where the measurements from the models track desired reference signals in the given time, of flexible beam models. The controller designs are based on the so-called internal model principle and they utilize difference between measurement and desired reference trajectory. Moreover, the controllers are robust in the sense that they can achieve output regulation despite external disturbances and model uncertainties. We also study the output regulation problem when there are certain limitations on the control input. In particular, we generalize the theory of output regulation for dynamical systems described by ordinary differential equations subject to input constraints to a particular class of systems described by partial differential equations. We present set of solvability conditions and a linear output feedback controller for the output regulation

Spectral analysis and Riesz basis property for vibrating systems with damping

In this thesis, we study one-dimensional wave and Euler-Bernoulli beam equations with Kelvin-Voigt damping, and one-dimensional wave equation with Boltzmann damping. The spectral property of equations with clamped boundary conditions and internal Kelvin-Voigt damping are considered. Under some assumptions on the coe±cients, it is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of ¯nite algebraic multiplicity, and the continuous spectrum that is identical to the essential spec- trum is an interval on the left real axis. The asymptotic behavior of eigenvalues is also presented. Two di®erent Boltzmann integrals that represent the memory of materials are consid- ered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the in¯nity, the spectrum of system contains a left half complex plane, which is sharp contrast to most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the later case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: There is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008