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

Tobias Breiten; Karl Kunisch

doi:10.3934/dcds.2020178

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2020, Volume 40, Issue 7: 4197-4229. Doi: 10.3934/dcds.2020178

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

Tobias Breiten^1, , and
Karl Kunisch^1,2,

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

Received: April 30, 2019

Revised: December 31, 2019

Published: April 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35, 49J20, 49N35, 93D05, 93D15.

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