具有混合边界条件的斯托克斯方程最优边界控制的存在性
Existence of optimal boundary control for the Navier-Stokes equations with mixed boundary conditions
摘要：
变分方法已成功地作为一种策略，以利用实际数据的测量。在一些应用中，这种方法提供了一种提高数值模拟精度的方法。在流体动力学的特殊情况下，它导致了具有非标准代价泛函的最优控制问题，当约束为Navier-斯托克斯方程时，需要一个非标准的理论框架来保证解的存在性。本文证明了一类此类最优控制问题解的存在性。在此之前，我们确保三维定常Navier-斯托克斯方程解的存在性和唯一性，混合边界条件，一种特殊类型的边界条件，在生物医学问题的应用中非常常见。
英文摘要：
Variational approaches have been used successfully as a strategy to take advantage from real data measurements. In several applications, this approach gives a means to increase the accuracy of numerical simulations. In the particular case of fluid dynamics, it leads to optimal control problems with non standard cost functionals which, when constraint to the Navier-Stokes equations, require a non-standard theoretical frame to ensure the existence of solution. In this work, we prove the existence of solution for a class of such type of optimal control problems. Before doing that, we ensure the existence and uniqueness of solution for the 3D stationary Navier-Stokes equations, with mixed-boundary conditions, a particular type of boundary conditions very common in applications to biomedical problems.
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