I am about to test my algorithms for solving optimal control problems of type:

Find an input $u$, such that for a time interval $(0,T]$ the cost functional $$J(v,u) = \mathcal M(v(T)) + \int_0^T\mathcal K(v(t),u(t)) dt, $$ becomes minimal, where $v$ is the velocity of flow modelled by the incompressible Navier-Stokes equations. The functionals $\mathcal M$ and $\mathcal K$ measure the state trajectory, e.g. in terms of how well a given target flow is approximated.

For 2D, typical test problems I have seen comprise the backward facing step, driven cavity (as in the pics below) or the cylinder wake. However, I haven't found discussions on

What is a good configuration, or even a benchmark, for testing numerical solution approaches to optimal flow control?

I want to do my first tests on a very basic level. That's why I think the setup should be

  • reasonably sized (~$10^5$ DOFs),
  • laminar,
  • and for distributed control.

Here, distributed means that the control appears in the right hand side of the Navier-Stokes equations and not in the boundary conditions.

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