摘要
本文研究半线性抛物型方程确定未知系数 a(x)的反问题.我们考虑泛函,其中 U(x,t,a)是初边值问题的解,U~*(x,t),a~*(x)是观测数据,寻求 a=a_0使得 J(a_0)≤J(a)对于任何元素 a 成立,这样的问题称为反问题的最优控制,利用能量不等式,证明了一个存在性定理.
In this paper,we deal with the inverse problem of determining the unknown coefficient a(xfor the problem _1U-(a(x_x_1U=f(x,t,U.We consider the cost functionJ(a=‖u(x,t,a-U(X,t‖_(L^2(Q_T+e‖a-a‖_(L^2(Q,where U(x,t,ais the so- lution of the problem for an instant function a(x,U(x,tand a(xare measurement.Find a=a_0 which minimizes J(athen,this problem is called the optimal control problem of the inverse problem.We will prove an existence theorem of the problem.
基金
This work was supported by the University Natural Science Foundation
关键词
最优控制
反问题
半线性抛物方程
存在定理
optimal control
inverse problem
semilinear parabolic equation
existence theorem
