Control and Stabilization of the Rosenau Equation Posed on a Periodic Domain

This paper considers the Rosenau equation with a moving control

$${\partial _t}u + {\partial _t}\partial _x^4u + {\partial _x}u + u{\partial _x}u = a(x + ct)h(x,t),c \ne 0,x \in T = \mathbb{R}/(2\pi \mathbb{Z}),t > 0$$

. The authors prove that the Rosenau equation with a moving control is locally exact controllable in H ^s (T) with s ≥ 0 and globally exponential stable in H ^s (T) with s ≥ 2. The two results nontrivially extend the work of (Rosier L and Zhang B Y, 2013) from the BBM equation to the Rosenau equation.