Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations
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In this talk, we analyze optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. The main question is: can optimal control change the behavior and the stability of state solution branches when steering the system towards a preferable desired profile? A general framework for nonlinear optimal control problem is presented in order to reconstruct each branch of optimal solutions, discussing in detail their stability properties. 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 the stability eigenvalue analysis of the controlled state. Finally, we propose reduced-order modeling as a tool to efficiently and reliably solve the parametric stability analysis of such optimal control systems.
