The Controllability Development Course Of Parabolic Equation

This summarize is about the Controllability Development Course of Parabolic Equation .The summarize is divided into three parts . The first part is the controllability of heat equation .We consider a constrained approximate controllability problem for the heat equation ,on the lack of null-controllability of the heat equation on the half-line ,and null exact controllability of the heat equation. The second part is the controllability of parabolic equation .We study on the controllability of parabolic systems with a nonlinear term involving the state and the gradient ,the null-controllability property of a 1-d parabolic equation involving a fractional power of the Laplace operator ,null controllability of some systems of two parabolic equations with one control force ,and exact controllability of the parabolic system with bilinear control .The third part is the controllability of semilinear heat equation. From a note on approximate controllability for one-dimensional heat equation ,some results on the controllability of linear and semilinear heat equation .