Volume 56, Number 4, July-August 2022
|Page(s)||1361 - 1400|
|Published online||27 June 2022|
Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction
mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy
2 Chair of Computational Mathematics and Simulation Science, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
3 Department of Mathematics and Physics, Catholic University of the Sacred Heart, Brescia, Italy
* Corresponding author: firstname.lastname@example.org
Accepted: 29 April 2022
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.
Mathematics Subject Classification: 35Q35 / 49J20 / 65P30
Key words: Parametrized nonlinear PDEs / optimal control problems / bifurcation analysis / Navier–Stokes equations / model order reduction
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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