Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain

In this paper, solutions of the third-order linear dispersion equations \[\frac{{\partial w}}{{\partial t}} + \frac{{\partial ^3 w}}{{\partial x^3 }} = f(x,t)\qquad {\text{and}}\qquad \frac{{\partial w}}{{\partial t}} + \frac{{\partial ^3 w}}{{\partial x^3 }} = 0\] are studied for $t \geqq 0$, $0 \leqq x \leqq 2\pi $. In the first case, periodic boundary conditions are imposed at 0 and $2\pi $ and the distributed control f, which may, however, have support smaller than $[0,2\pi ]$, is assumed to be generated by a linear feedback law conserving the “volume” $\int_0^{2\pi } {w(x,t)dx} $ while monotonically reducing $\int_0^{2\pi } {w(x,t)^2 dx} $. For the second equation, a feedback boundary control having the same properties is applied. In both cases, uniform exponential decay to a constant state is obtained. Related exact controllability questions are also studied, and affirmative results are obtained.