The recovery of parameters of a Navier–Stokes system

We study inverse problems about the recovery of a priori unknown parameters of a dynamic system described by a boundary value problem for a system of Navier–Stokes equations on the basis of some available information on the motion of the dynamic system (on the solution of a corresponding boundary value problem). The first of the problems under consideration is a retrospective inverse problem, consisting in the recovery of a priori unknown initial state of a dynamic system on the basis of its known final state. This problem, as many other similar problems, is ill-posed. For its solution, ideas of the so-called start control are involved. Realization of these ideas allows one to reduce the original ill-posed problem to a set of direct well-posed problems which are solved in the direct time under corresponding given initial conditions. We also consider various modifications and regularizations of the suggested method of solution of the problem based on some a priori information about a desired solution. The second of the problems under discussion is an inverse dynamic problem, consisting in the dynamic recovery of a priori unknown right side of the system which characterizes the density of external mass forces, by results of approximate measurements of current system states. This problem is also ill-posed. For its solution we suggest a dynamic positional regularizing algorithm physically realizable and able to work in a real-time mode. Construction of the algorithm is supported by structures and methods of the positional control theory and methods of regularization of ill-posed problems.