The Research Summary Of The Controllability Of Semilinear Heat Equation

Heat equations is a kind of fundamental equations in the controllability of PDES.In recent years,the control problems of the semilinear heat equations has received a great deal of attention from mathematicians.In this article we have done some research sum-mary into the controllalility for semilinear heat equations. First,we introduce the con-trollability for the linear heat equations in bouded domains of R. Then,we consider the approximate controllability for the semilinear heat equation. Approximate control-lability for the semilinear heat equation is got by this way that linearing the system by variational approach and constructing functional and using dual equation. At last, using the Browder-Minty theorem from the theory of monotone operators,we consider the exact internal controllability for the semilinear heat equation.We show that the system is exactly controllable in L~2(Ω) if the nonlinearities are globally Lipschitz con-tinuous.