# Computational Mathematics and Scientific Computing Seminar

## Numerical Aspects of Flow Stabilization by Riccati Feedback

**Speaker:**
Heiko Weichelt, The Mathworks, UK

**Location:**
Warren Weaver Hall 1302

**Date:**
May 12, 2017, 10 a.m.

**Synopsis:**

Various transport and flow problems are important in many technical applications. To control or influence these generally nonlinear problems, one usually uses an open-loop controller. Highly sophisticated solvers exist for this kind of problem. However, an open-loop controller is unstable with regard to small perturbations. Using a feedback stabilization approach for the linearization around the open-loop trajectory increases the robustness of these methods drastically. The main issue in deriving such a Riccati-based feedback is the efficient solution of a large-scale generalized algebraic Riccati equation. The solution of this quadratic matrix equation is derived by applying a specially tailored version of Newton’s method. Due to the natural divergence-free condition of incompressible flow problems, all systems occur as differential-algebraic systems. The solution of this kind of equations is highly demanding. We try to avoid general-purpose solution strategies for these differential-algebraic equations by using and extending an existing implicit projection method. A highly efficient algorithm to determine the Riccati-based feedback for various flow scenarios has been established by modifying and combining various existing solution strategies. The combination of all these strategies is completely new and only possible because of current improvements. The key ingredient to enable the synergy of these methods are low-rank structures and specially tailored algorithms that exploit these structures. A convergence proof for our proposed method as well as thorough numerical experiments verify the usability of our approach. By modifying and extending an existing finite element flow solver, we have been able to extract the arising finite dimensional matrices. These matrices are used within our MATLAB-based algorithms to compute the Riccati-based feedback. The computed feedback is included into the flow solver such that a closed-loop simulation is able to validate the usability of our proposed approach to stabilize the Navier–Stokes equations over the Kármán vortex street.