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.
Keywords: Parametrized nonlinear PDEs, optimal control problems, bifurcation analysis, Navier–Stokes equations, model order reduction
@article{M2AN_2022__56_4_1361_0,
author = {Pichi, Federico and Strazzullo, Maria and Ballarin, Francesco and Rozza, Gianluigi},
title = {Driving bifurcating parametrized nonlinear {PDEs} by optimal control strategies: application to {Navier{\textendash}Stokes} equations with model order reduction},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1361--1400},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {4},
doi = {10.1051/m2an/2022044},
mrnumber = {4444527},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022044/}
}
TY - JOUR AU - Pichi, Federico AU - Strazzullo, Maria AU - Ballarin, Francesco AU - Rozza, Gianluigi TI - Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1361 EP - 1400 VL - 56 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022044/ DO - 10.1051/m2an/2022044 LA - en ID - M2AN_2022__56_4_1361_0 ER -
%0 Journal Article %A Pichi, Federico %A Strazzullo, Maria %A Ballarin, Francesco %A Rozza, Gianluigi %T Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1361-1400 %V 56 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022044/ %R 10.1051/m2an/2022044 %G en %F M2AN_2022__56_4_1361_0
Pichi, Federico; Strazzullo, Maria; Ballarin, Francesco; Rozza, Gianluigi. Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1361-1400. doi: 10.1051/m2an/2022044
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**Both authors contributed equally to this work.
