Approximate Controllability Of The Semilinear Elliptic Equation

In this paper,we systematically study the approximate controllability for the semi-linear elliptic equation in a bounded domain Ω,when the control acts on any open and nonempty subset of Ω.With the help of the classical Fenchel-Rockafellar's duality theory,we transform the semilinear elliptic equation to the linear elliptic equation. By proving the unique existence of the minimal element of the dual function,we obtain the unique existence of the minimal element of the original function.Then we gain the L~p-approximate controllability(1 < p < ∞) of the linear elliptic equation.In the following, by employing the. Kakutani's fixed point theory,we transform the L~p-approximate controllability of the linear elliptic equation problem to the L~p-approximate controllability of the semilinear elliptic equation problem. Consequently,we get the main results of this paper.Meanwhile, we simply study the approximate controllability of this system in L~1(Ω) and c~0(Ω), which generalize and improve related studies in the literature,at the same time.we attain some new results.