# American Institute of Mathematical Sciences

July  2020, 40(7): 4197-4229. doi: 10.3934/dcds.2020178

## Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations

 1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria 2 RICAM Institute, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria

* Corresponding author: tobias.breiten@uni-graz.at

Received  May 2019 Revised  January 2020 Published  April 2020

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.

Citation: Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178
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