Exact Controllability For A Wave Equation With Moving Boundary In A Non-cylindrical Domain

It is necessary to study the exact controllability of partial differential equations,Because this theory plays a key role in the research of relevant theoretical and practical application.The main work of this paper is to study the exact controllability of wave equation with mixed moving boundary in non-cylindrical region.This paper has three chapters.The first chapter is the introduction part,which mainly introduces some related documents and the main conclusions of this article.In the second chapter,we mainly consider the wave equation in a non-cylindrical region(?)where (?) is the state variable,(?) is the control variable and(?)is any given initial value.In the second chapter,firstly,multipliers are directly selected on non-cylindrical regions.Then,the energy of the dual system is found and its attenuation is obtained.Finally,through HUM,the exact controllability of the original system is obtained and the control time is given.In particular,this time is a smaller control time.In the third chapter,we consider a string equation with mixed boundary in the noncylindrical domain(?)where (?) is the state variable, is the control variable and(?)is any given initial value.Similar to the second chapter,Firstly,the appropriate multiplier is directly selected on the non-cylindrical region,then the energy attenuation of the dual system is deduced,and finally the exact controllability of the original system is obtained by HUM.