We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α→∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

中文翻译：

---

具有Navier滑移边界条件的粘性不可压缩流体最优控制问题的渐近分析。

我们考虑带有Navier滑移边界条件的Navier–Stokes系统的最优控制问题。我们用α表示摩擦系数，并分析α→∞这样的问题的渐近行为。更确切地说，我们证明，如果对每个α采取最优控制，则存在一系列最优控制，它们收敛到具有Dirichlet边界条件的Navier-Stokes系统的相同最优控制问题的最优控制。我们还显示了相应的直接状态和伴随状态的收敛性。