Time-independent feedbacks for the null controllability of homogeneous quasilinear hyperbolic systems in one dimensional space

We consider the null controllability of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks for the null-controllability for $C^1$-solutions at any time larger than the optimal time for the linear system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.