Inverse problems of reconstructing parameters of the Navier-Stokes system
An inverse problem of reconstructing parameters not known a priori of the dynamical system described by the boundary-value problem for the Navier-Stokes system is considered. The reconstruction is based on one piece of admissible information or another about the motion of the dynamical system (solution of the corresponding boundary-value problem). In particular, one of the problems considered is the inverse problem to reconstruct the a priori unknown right-hand side of the Navier-Stokes system. The right-hand side characterizes the density of exterior mass forces acting on the system. This problem, as well as many other similar problems, is ill-posed. Two methods are proposed for its solution: the statistical method and the dynamical method. These methods use different initial information. In solving the problem by using the statistical method, initial information for the solution is the approximate measurement (in one sense or another) of the motion of the dynamical system on a given interval of time. Here, the reconstruction is performed after the corresponding interval of time. For the solution of the problem by this method open-loop control theory are used. In solving the problem by using the dynamical method, initial information for its solution is the approximate (in one sense or another) measurements of the current states of the system, which are dynamically obtained by the observer. Here, the reconstruction is dynamically performed during the process. For the solution of the problem by the dynamical method, the dynamical regularization method based on positional control theory are used. Also, the author considers various modifications of the methods proposed that are based on a priori information about the desired solution and solvability conditions of the problem.
scientific article; zbMATH DE number 5147493
scientific article