论文标题
通过最佳控制策略驱动分叉的参数化非线性PDE：应用于减少模型订单的Navier-Stokes方程
Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction
论文作者
论文摘要
这项工作将最佳控制问题作为一种策略，以将非线性参数化偏微分方程的分叉解决方案推向所需的分支。实际上，对于这些管理方程式，可以由相同的参数实例产生多个解决方案配置。因此，我们旨在描述最佳控制如何更改解决方案轮廓和状态解决方案分支的稳定性。首先，提出了非线性最佳控制问题的一般框架，以重建最佳解决方案的每个分支，详细讨论所获得的受控解决方案的稳定性。然后，我们将提出的框架应用于突然暴露通道中的Navier-Stokes方程来控制的几个最佳控制问题，描述了控制对干草叉分叉的定性和定量效应，并详细评论了受控状态的稳定性特征分析。最后，我们建议减少订单建模作为一种工具，以有效且可靠地求解此类最佳控制系统的参数稳定性分析，这对于使用标准离散化技术（例如有限元方法）可能具有挑战性。
This work deals with optimal control problems as a strategy to drive bifurcating solution of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches. First of all, a general framework for nonlinear optimal control problem is presented in order to reconstruct each branch of optimal solutions, discussing in detail the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations in a sudden-expansion channel, describing the qualitative and quantitative effect of the control over a pitchfork bifurcation, and commenting in detail the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which can be challenging to perform with standard discretization techniques such as Finite Element Method.
