Lyapunov exponents and control sets near singular points

We consider control affine systems of the form on with compact control range U and a singular point , i.e. fi(x*)=0, i=0,…,m. Control sets are maximal subsets of with nonvoid interior where the system is approximately controllable. We suppose that there is a periodic control functions uh such that linearized the system has positive and negative Lyapunov exponents for uh and a periodic control function us such that the linearized system has only negative Lyapunov exponents for us. Under an inner pair condition we show the existence of control sets near the singular point, by using local stable and unstable manifolds of the system corresponding to uh and its asymptotic phase.