Effect Of The Diffusive Coefficient In The Controllability Of The Linear Parabolic System

In this paper we are concerned with estimates of the cost of approximate controllability and null controllability of parabolic equations in divergence form.Especially,the explicit bound of the cost with respect to the diffusive coefficient a(x,t) and the time T is analyzed. As an application,we prove the approximate controllability to a kind of quasilinear parabolic equations(with nonlinear principle part).The methods of proof combines global Carleman inequality,energy estimates for parabolic equations,the variational approach to controllability and the Kakutani's fixed point theorem.