ABSTRACT

Optimal control of nonstationary Navier–Stokes equations is studied with nonlinear boundary conditions described by the Clarke subdifferential. Precisely, we aim at minimizing a general functional for a control problem whose state is a solution to a boundary value problem depending on the control itself. Accordingly, the lower level problem is expressed by a hemivariational inequality associated with a nonconvex nonsmooth locally Lipschitz superpotential. The existence of solutions to our problem is then shown via a convergence scheme based on mixed equilibria and a stability result with respect to variations on the control for the dynamic state control system associated with the main control problem.

摘要

用克拉克次微分描述的非线性边界条件研究了非平稳 Navier-Stokes 方程的最优控制。准确地说，我们的目标是最小化控制问题的一般功能，其状态是取决于控制本身的边界值问题的解决方案。因此，较低级别的问题由与非凸非光滑局部 Lipschitz 超电势相关的半变分不等式表示。然后通过基于混合平衡的收敛方案和与主控制问题相关的动态控制系统的控制变化的稳定性结果来显示我们问题的解决方案的存在。