The Design Of Optimal Position Control On The One-dimensional Heat Equation

In this paper , we discusses the controllability of right-hand side with the Dirichletboundary condition with a one-dimensional heat equation the value of the optimal locationof control. Firstly, we proved the existence and uniqueness of weak solutions of the stateequation by using Galerkin approximation method; Secondly, using the solution of adjointequation to estimate uh and obtain the estimator inequality ,then gives the optimal positionto meet the necessary conditions; Finally, a special case of this paper to verify the necessaryconditions. This article is divided into three parts, the first introduced to main problem andsome preliminaries. The second part is the proof of the existence and uniqueness of weaksolution of the one-dimensional heat equation. The third part discusses the existence anduniqueness of the optimal position control of and the necessary conditions of the optimalposition control, and then the special case is given to verify the above necessary conditions.