The purpose of this article is to introduce radial basis function (RBF) methods for solving both direct Stokes equations and controllability problems for the Stokes system with few internal scalar controls. In both cases, Dirichlet or Navier-slip boundary conditions are considered. We introduce two radial basis function solvers, one global and the other local, to solve Stokes equations. These methods are used to discretize the primal and adjoint systems related to the controllability problems. Both techniques are based on divergence-free global RBFs. A global colocation technique based on Div-free inverse multi-quadrics is formulated and analyzed. A generalization of scalar hybrid kernels to a vector divergence-free hybrid RBFs setting is defined. Based on these kernels, the local Hermite interpolation (LHI) method in vector form is introduced. Due to the properties of the hybrid kernel, we show that due to the properties of the hybrid kernel this local method, can reduce up to double precision, the value of the condition number of the local Gram matrices. Simultaneously, it is proved that the real components of the eigenvalues corresponding to the global LHI matrix are negative and that consequently backward difference formulas are stable for time integration. The conjugate gradient algorithm is adapted to the radial basis function setting to solve the controllability problems. Several benchmarks problems in two dimensions with a non-convex domain (a star shape) are numerically solved by these RBFs methods to display and compare their feasibility. The solutions to these problems are also implemented by finite element techniques to study their relative performance.

本文的目的是介绍径向基函数（RBF）方法，以解决内部标量控制很少的Stokes系统的直接Stokes方程和可控性问题。在两种情况下，都考虑Dirichlet或Navier滑移边界条件。我们引入了两个径向基函数求解器（一个是全局的，另一个是局部的）来求解Stokes方程。这些方法用于离散化与可控性问题有关的原始和伴随系统。两种技术均基于无散度的全局RBF。提出并分析了一种基于Div-free逆多二次方的全局代管技术。定义了标量混合内核到无矢量散度混合RBF设置的一般化。基于这些内核，介绍了矢量形式的局部Hermite插值（LHI）方法。由于混合核的特性，我们证明了由于混合核的特性，该局部方法可以降低高达两倍精度，即局部Gram矩阵的条件数的值。同时，证明了与全局LHI矩阵相对应的特征值的实分量为负，因此后向差分公式对于时间积分是稳定的。共轭梯度算法适用于径向基函数设置，以解决可控性问题。通过这些RBF方法在数值上解决了非凸域（星形）二维的几个基准问题，以显示和比较它们的可行性。这些问题的解决方案也可以通过有限元技术来实现，以研究其相对性能。