Controllability For Heat Equations With Constrained Control

So far, great progresses have been made in the field of controllability without constraint.It is meaningful to study controllability with constrained control because of requirement of theory and practical problems. The study of constrained controllability theory, even for the linear system, is still at its infant stage.This paper is concerned with controllability for heat equations with bounded control.Heat equations with Dirichlet and Neumann boundary conditions are discussed, separately. Apparently, the approximate controllability doesn't hold for generally linear parabolic equations with bounded control. It doesn't hold for the semilinear heat equations presented there,either. Then, it is proved by analyzing its' solutions that no matter how long the time T is , there are targets that can't be approximated by the solution of heat equation with Dirichlet boundary conditions;If a given target satisying certain conditions can be reached by the solution of heat equation . At last, it is proved for the two systems that when the control domain U is a bounded, closed set, the reachable set is the same as that when the control domain is the convex of U.It is an advancement for original results.