RBFs methods for null control problems of the Stokes system with Dirichlet and Navier slip boundary conditions

The purpose of this article is to introduce radial basis function, (RBFs), methods for solving null control problems for the Stokes system with few internal scalar controls and Dirichlet or Navier slip boundary conditions. To the best of our knowledge, it has not been reported in the literature any numerical approximation through RBFs to solve the direct Stokes problem with Navier slip boundary conditions. In this paper we fill this gap to show its application for solving the null control problem for the Stokes system. To achieve this goal, we introduce two radial basis function solvers, one global and the other local, to discretized the primal and adjoint systems related to the control problem. Both techniques are based on divergence free global RBFs. Stability analysis for these methods is performed in terms of the spectral radius of the corresponding Gram matrices. By using a conjugate gradient algorithm, adapted to the radial basis function setting, the control problem is solved. Several test problems in two dimensions are numerically solved by these RBFs methods to test their feasibility. The solutions to these problems are also obtained by finite elements techniques, (FE), to compare their relative performance.