论文标题
对具有控制约束的3D阻尼Navier-Stokes-Voigt方程的最佳控制
Optimal Control of the 3D Damped Navier-Stokes-Voigt Equations with Control Constraints
论文作者
论文摘要
在本文中，我们考虑了具有非线性阻尼$ | u | u |^{r-1} u，in [1，\ infty）$中的3D Navier-Stokes-voigt（NSV）方程。我们制定了一个最佳的控制问题，可以最大程度地减少能量标准中速度场的卷曲，但要以分布式控制力满足阻尼的NSV方程的流速度。该控件还需要遵守盒子型约束。对于任何$ r \ geq 1，$ $在$ \ mathbb r^3 $中定期/界定域$ω$时，讨论了弱解决方案的存在和唯一性，而在空间周期边界条件的情况下则获得了独特的强解决方案。我们证明了用于控制问题的最佳对的存在。使用经典的伴随问题方法，我们表明，最佳控制满足了各种不等式给出的一阶必要最佳条件。由于最佳控制问题是非凸的，因此我们获得了二阶足够的最佳条件，表明可允许的控制在本地是最佳的。此外，我们根据与全球最佳控制的阻尼项的增长定义的伴随状态得出最佳条件。
In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping $|u|^{r-1}u, r\in[1,\infty)$ in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any $r\geq 1,$ the existence and uniqueness of a weak solution is discussed when the domain $Ω$ is periodic/bounded in $\mathbb R^3$ while a unique strong solution is obtained in the case of space-periodic boundary conditions. We prove the existence of an optimal pair for the control problem. Using the classical adjoint problem approach, we show that the optimal control satisfies a first-order necessary optimality condition given by a variational inequality. Since the optimal control problem is non-convex, we obtain a second-order sufficient optimality condition showing that an admissible control is locally optimal. Further, we derive optimality conditions in terms of adjoint state defined with respect to the growth of the damping term for a global optimal control.
