The Study On Control And Stability Of Networks Of Euler-Bernoulli Beams

This thesis studies the control and stability analysis of network described by Euler-Bernoulli beam equations. We focus on the design of controllers and the analysis in thespectrum of closed loop system operator. Consequently, we discuss the asymptoticalstability and exponential stability of the system.A pinned network of Euler-Bernoulli beams is studied. Based on the energy dis-sipation method, we design the feedback controllers for the system and then prove thatthe closed loop system is well posed. We show further that the root vectors of the sys-tem operator are complete, and there exist a sequence of the root vectors that form aRiesz basis for the state space. By detailed analysis we get the asymptotic spectrumof the system. Note that the Riesz basis property implies that the system satisfies thespectrum determined growth condition. Then we prove that the imaginary axis is notthe asymptote of the eigenvalues under some conditions. Hence the system is exponen-tially stable under some conditions..The next model being studied is a simple triangle network of Euler-Bernoullibeams.We design the feedback controllers for the network and then prove that theclosed loop system is well posed. We prove that the imaginary axis is not the asymp-tote of the eigenvalues, which means that the system is asymptotically stable. We showfurther that even we hold on the two controllers at the common node, the system wouldno longer be stable, if we take one beam away from the triangle network. By detailedanalysis we get the asymptotic spectrum of the system. The completeness of the rootvectors of the system operator is proved, that is, there exist a sequence of the rootvectors that form a Riesz basis for the state space. Note that the Riesz basis propertyimplies that the system satisfies the spectrum determined growth condition. Hence thesystem is exponentially stable.The general network of beams are studied. The first model is a network withfive Euler-Bernoulli beams. We design the feedback controllers for the network of Euler-Bernoulli beams and then prove that the closed loop system is well posed. Weshow further that the root vectors of the system operator are complete, and there exista sequence of the root vectors that form a Riesz basis for the state space. By detailedanalysis we get the asymptotic stability of the system. Note that the Riesz basis prop-erty implies that the system satisfies the spectrum determined growth condition. Hencethe system is exponentially stable if and only if the imaginary axis is not the asymptoteof the eigenvalues. The next network of beams we studied has many loops. In addi-tion, there are edges outside these loops. The conditions for asymptotic stability of thesystem is studied.