Some recent results on feedback regularization of Navier-Stokes equations

We present several results from our work (1997) concerning feedback control of the three-dimensional Navier-Stokes system on a bounded domain with non-slip boundary conditions. The feedback is introduced in the form of an additional nonlinear viscosity term which is nonvanishing only on a subdomain of the main domain and is proportional to the energy dissipation functional on this subdomain. For the controlled system, in the above article we prove the existence, uniqueness and stability of the strong solution for initial data and forcing term which are arbitrary on the subdomain of control and are sufficiently small (in appropriate function spaces) outside this subdomain