  Feedback control synthesis for nonlinear, parameter-dependent fluid flow
control problems is considered. The optimal feedback law requires the solution
of the Hamilton-Jacobi-Bellman (HJB) PDE suffering the curse of dimensionality.
This is mitigated by Model Order Reduction (MOR) techniques, where the system
is projected onto a lower-dimensional subspace, over which the feedback
synthesis becomes feasible. However, existing MOR methods assume at least one
relaxation of generality, that is, the system should be linear, or stable, or
deterministic.
  We propose a MOR method called Statistical POD (SPOD), which is inspired by
the Proper Orthogonal Decomposition (POD), but extends to more general systems.
Random samples of the original dynamical system are drawn, treating time and
initial condition as random variables similarly to possible parameters in the
model, and employing a stabilizing closed-loop control. The reduced subspace is
chosen to minimize the empirical risk, which is shown to estimate the expected
risk of the MOR solution with respect to the distribution of all possible
outcomes of the controlled system. This reduced model is then used to compute a
surrogate of the feedback control function in the Tensor Train (TT) format that
is computationally fast to evaluate online. Using unstable Burgers' and
Navier-Stokes equations, it is shown that the SPOD control is more accurate
than Linear Quadratic Regulator or optimal control derived from a model reduced
onto the standard POD basis, and faster than the direct optimal control of the
original system.
