Output Feedback Stabilization Of Parallel Connected String Equations

With the development of science technology,vibration controls of such system have received much attention.This thesis studies the given two parallel connected string equations which divided into three chapters.In the first chapter,we introduce the background and research on this field as well as give some definitions and basic lemmas.In the second chapter,we consider the following system utt(x,t)= uxx(x,t)+ ?(v(x,t)—u(x,t)),(x,t)?(0,1))×((0,+?),vtt(x,t)= vxx(x,t)+ ?(u(x,t)—v(x,t)),(x,t)?(0,1)x(0,+?),ux(0,t)? 0,t?0,vx(0,t)= 0,t?0,ux(1,t)=-?ut(1,t)-?u(1,t),t ? 0,vx(1,t)=-?vt(1,t)-?v(1,t),t ? 0,where t and x characterized for time and space variable,u(x,t)and v(x,t)represent the displacement of two string in position x at timet,?>0 is the internal parameter of the couple strings,?,?>0 are system boundary parameters.This system describes the vibration behavior of the internal coupled parallel system.We choose the proper state spaceH,define the operator of system,using the inner product theorem we prove that the operator of system is dissipative.And next we use the Lumer-philips's theorem to prove that A generate a compress Co-semigroup eAt,so we solve the problem of solution existence.But the method of Lyapunov cannot be applied in the stability of our system,so we have to use the Huang's theorem,through the following estimate M = sup{||(A-i?)-1||H|??R}<?,we finally proved that the system is exponential stable.In this chapter,we are concerned with boundary output feedback stabilization of two parallel connected string equations with matched external disturbance:utt(x,t)-uxx(x,t)= ?(v(x,t)-(x,t)),x ?(0,1),t>0,vtt(x,t)-vxx(x,t)= ?(u(x,t)-v(x,t)),x ?(0,1),t>0,u(0,t)= 0,t ? 0,ux(1,t)= U1,(t)+ d1(t),t ?0,v(0,t)= 0,t ?0 vx(1,t)= U2(t)+ d2(t),t?0,u(x,0)? u0(x),ut(x,0)= u1(x),x ?[0,1],v(x,0)= vO(x),vt(x,0)v1(x),x?[0,1],y(t)= {ux(0,t),vx(1,t),u(1,t),v(1,t),ut(1,t),vt(1,t)}where(u,v)T is the state,Ui,U2 are the input(control),y(t)is output,?>0 is a constant parameter,d1,d2 are unknown disturbance.By the method proposed in[7],we design a disturbance estimator first which is used to cancel effect of disturbance in the feedback loop.The other part of output feedback controller is designed to make system stable.Then operator semigroup and Lyapunov function methods are adopted to prove that the closed-loop system admits a unique solution in state space,the solution of original system is asymptotically stable and the one of the disturbance estimator system is bounded.Finally,numerical simulations are presented to validate the theoretical results.