Exact controllability of nodal profile for 1-D first order quasilinear hyperbolic systems with internal controls

For 1-D first order quasilinear hyperbolic systems without zero eigenvalues, based on the theory of exact boundary controllability of nodal profile, using an extension method, the exact controllability of nodal profile can be realized in a shorter time by means of additional internal controls acting on suitably small space-time domains. On the other hand, using a perturbation method, the exact controllability of nodal profile for 1-D first order quasilinear hyperbolic systems with zero eigenvalues can be realized by additional internal controls to the part of equations corresponding to zero eigenvalues. Furthermore, by adding suitable internal controls to all the equations on suitable domains, the exact controllability of nodal profile for systems with zero eigenvalues can be realized in a shorter time.