Reduced order framework for optimal control of nonlinear partial differential equations: ROM-based optimal flow control

A variety of partial differential equations (PDE) can govern the spatial and time evolution of fluid flows; however, direct numerical simulation (DNS) of the Euler or Navier-Stokes equation or other traditional computational fluid dynamics (CFD) models can be computationally expensive and intractable. An alternative is to use model order reduction techniques, e.g., reduced order models (ROM) via proper orthogonal decomposition (POD) or dynamic mode decomposition (DMD), to reduce the dimensionality of these nonlinear dynamical systems while still retaining the essential physics. The objective of this work is to design a reduced order numerical framework for effective simulation and control of complex flow phenomena. To build our computational method with this philosophy, we first simulate the 1D Burgers' equation ut + uux ? ?uxx = f(x, t), a well-known PDE modeling nonlinear advection-diffusion flow physics and shock waves, as a full order high resolution benchmark. We then apply canonical reduction approaches incorporating Fourier and POD modes with a Galerkin projection to approximate the solution to the posed initial boundary value problem. The control objective is simple: we seek the optimal (pointwise) input into the system that forces the spatial evolution of the PDE solution to converge to a preselected target state uT(x) at some final time T > 0. To implement an iterative control loop, we parametrize the unknown control function as a truncated Fourier series defined via a set of finite parameters. The performance of the POD ROM is compared to that of the Fourier ROM and full order model for six numerical experiments