Null Controllability Of A Class Of Coupled System

This thesis studies the following null controllability of a coupled system（?）where u（x,0）=u₀（x）,v（x,0）=v0（x）,and u₀（x）∈ H²ï¼ˆI）∩ H01（I）,v0（x）∈L²ï¼ˆI）.The control is h ∈ L²ï¼ˆQ）,which only acts on the heat equation.Firstly,we study the linear coupled system.Our method for investigating the null controllability of the system is based on the energy estimates,projective theorem and regularity of parabolic equation.We shall prove that for every u₀∈L²ï¼ˆI）,v0∈L²ï¼ˆI）and a=?²-1,a∈M’,d∈[δ,+∞),the solutions of the coupled system satisfy u（x,T,h）=v（x,T,h）=0,x ∈I.And h satisfies ‖h‖L²ï¼ˆQ）≤ C（‖u₀‖H²ï¼ˆI）∩H01（I）+‖v0‖L²ï¼ˆI））.Then,using Kakutani fixed point theorem and the results of the linear coupled system,we study the null controllability result for the one-dimensional coupled system.