Approximate controllability of the nonlinear third-order dispersion equation

In this paper, we consider the approximate controllability of nonlinear third-order dispersion equation of the form∂w∂t(x,t)+∂3w∂x3(x,t)=(Gu)(x,t)+f(t,w(x,t))on the interval 0 ⩽ x ⩽ 2π, t ⩾ 0 with initial and periodic boundary conditionsw(x,0)=0,∂kw∂xk(0,t)=∂kw∂xk(2π,t),k=0,1,2.We study the approximate controllability for nonlinear dispersion system under the assumption that the corresponding linear control system is approximately controllable. The solutions are given by a variation of constants formula which allows us to study the approximate controllability for nonlinear dispersion systems. Based on the semigroup theory and fixed point approach, sufficient conditions are formulated and proved.