Mathematics

 

This talk deals with optimal control problems as a strategy to drive bifurcating

solutions 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 the solution to change profile and the stability of state solution branches. 

We present a general framework for nonlinear optimal control problems 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, 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.

 

This is a Joint work with Federico Pichi, Francesco Ballarin and Prof. Gianluigi Rozza.