Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction

Abstract

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.

References in corpus (13)

• Model Reduction For Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering
• A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions
• Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology
• On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics
• Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation
• Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method
• Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation
• Emergence and Stability of Vortex Clusters in Bose-Einstein Condensates: a Bifurcation Approach near the Linear Limit
• Model order reduction for bifurcating phenomena in Fluid-Structure Interaction problems
• Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences
• Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature
• POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations
• Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations

Cited by in corpus (8)

• An artificial neural network approach to bifurcating phenomena in computational fluid dynamics
• A local basis approximation approach for nonlinear parametric model order reduction
• An optimisation-based domain-decomposition reduced order model for the incompressible Navier-Stokes equations
• An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems
• Control of bifurcation structures using shape optimization
• POD-based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control
• Sparse Identification for bifurcating phenomena in Computational Fluid Dynamics
• Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs