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 unbearable to perform with standard discretization techniques such as Finite Element Method.

Model order reduction for bifurcating phenomena in fluid‐structure interaction problems

“…Despite this, since the high‐fidelity models usually require unaffordable computational costs due to fine mesh discretization and repeated evaluations, we also focused on the application of reduced order models (ROM) 12‐17 . In particular, we employed the Reduced Basis (RB) method, 18,19 which has been already used to trace the bifurcating behavior of many problems in fluid‐dynamics, 20‐23 with further applications to other fields such as quantum mechanics, 24 computational mechanics 25 and bio‐mathematics 26 . This method is based on projecting the nonlinear governing equations on a reduced space of much lower dimension.…”