Exact Controllability For Wave Equations In Non-cylindrical Domains In One Dimension

The exact controllability for partial differential equation is an important research topicin the control theory,which has important theoretical significance and application value.This paper is devoted to the study on the exact controllability for wave equations in non-cylindrical domains.The paper consists of three chapters.In Chapter 1,we introduce the research results of the related work and the main resultsobtained in this paper.In Chapter 2,we consider a string equation in the non-cylindrical domain QkT:where u is the state variable,(u0,u1)?L2(0,1)×H-1(0,1)is any given initial data and vis the control variable.By using the multiplier method directly in non-cylindrical domains,we obtain the decay estimates for the energy of the dual system,thus we can get the exactboundary controllability for the original system.Also,we get a controllability time which issmaller than[13].In Chapter 3,we consider a wave equation with variable coefficient in the non-cylindricaldomain QkT:where a(x)?C1(0,1)and satisfies a(x)?1.u is the state variable,(u0,u1)?L2(0,1)xH-1(0,1)is any given initial data and v is the control variable.By using the multipliermethod directly in non-cylindrical domains,we obtain the decay estimates for the energy ofthe dual system.