Control Design For Disturbance Or Delay Rejection And Polynomial Stability Analysis For Elastic Systems

CDPS,the control of distributed parameter systems,is indisputably at the leading edge of the field of mathematical control theory.Since its inception in the 1960 s,along with modern control theory,the field of CDPS has undergone a burst of growth since the 1970 s,and rapid developments sine the 1980’s.The study of CDPS mainly contains control designs and system analysis.On the one hand,due to extensive applications of the system with disturbance or delay,the study of anti-disturbance and anti-delay con-troller Design becomes a hotspot research.On the other hand,more and more scholars devote to study the asymptotic stability,especially polynomial stability,of DPS,due to limitations of the model itself or controlled conditions,systems are often lack of expo-nential decay.Hence It is urgent to study how to design reasonably controllers to cancel disturbances or reject time delays and find a simple and feasible polynomial stability criterion.The objective of the thesis is to provide new methods to study how to design anti-disturbance or anti-delay controllers and how to verify the polynomial stability of a system.This article contents specifically as follows:1.In terms of anti-disturbance controllers,we consider the stabilization problem of an Euler-Bernoulli beam with tip mass,which undergoes nonuniform or uniform bounded disturbance,respectively.Using the above mentioned design method of anti-disturbance controller,we design a feedback controller based on a disturbance estima-tor and nonlinear feedback control law,respectively.We apply the semigroup theory and the maximal monotone operator theory to prove the well-posedness of the result-ing closed system,respectively.We prove the exponential stability of the closed loop systems by the Lyapunov function approach.2.In terms of anti-delay controllers,we consider the feedback stabilization of an Euler-Bernoulli beam with the boundary time-delay disturbance.Due to unknown time-delay coefficient,the system might be exponentially increasing at the lack of con-trol.We design the feedback control law based on Lyapunov function method.Dif-ferent from usual use of Lyapunov function method,our approach is to combine the construction of Lyapunov functionals with the controller design,which will guarantee the system engergy function decays expnentially.In this procedure,we deduce the in-equality equations satisfied by the system parameters.We proved the well-posedness of the corresponding closed-loop system by using semigroup theory and the inequality equations is solvable.Moreover,the exponential decay rate of the system is estimated.3.In terms of the asymptotic stability analysis,We characterize the polynomial stability of Co semigroup T(t)with generator A on Hilbert space H at first.Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for H.By the asymptotic relation of the real part and imaginary part of eigenvalues of A,we give the optimal decay rate of polynomial stability of T(t).Moreover,we give the zeros distribution of certain equations.As an application,we il-lustrate our general results by an acoustical system.At last the asymptotic behavior of a coupled string-beam system and hyperbolic-parabolic coupled systems on some special networks are discussed.By detailed spectral analysis,the asymptotic expressions of the spectrum of these two systems.Then by the frequency domain approach together with the asymptotic expressions of the spectrum,we derive the optimal polynomial decay rate of the systems.