Approximate controllability for the μ-version of the b-family Camassa–Holm equations with a finite-dimensional force

Abstract

Considered herein is the approximate controllability of the $\mu$-$b$-Camassa–Holm equation on the circle. We first prove the well-posedness and stability of the $\mu$-$b$-family Camassa–Holm equations with a source term. Then we establish the asymptotic property of this equation. Finally, based on the Agrachev–Sarychev approach in geometric control theory, the approximate controllability for the $\mu$-version of the $b$-family Camassa–Holm equations with two dimensional external force is shown.