  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.

Numerical Analysis;Numerical Analysis