Feedback Stabilization of the Two-Dimensional Navier–Stokes Equations by Value Function Approximation

Tobias Breiten (1) , Karl Kunisch (1, 2) , Laurent Pfeiffer (3, 4)

1 IM - Institut für Mathematik [Graz]
2 RICAM - Johann Radon Institute for Computational and Applied Mathematics
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 COMMANDS - Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires

The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established arounda steady state, moreover, its derivatives are shown to satisfy a Riccati equation at theorder two and generalized Lyapunov equations at the higher orders. An approximationof the optimal feedback law is then derived from the Taylor expansion of the valuefunction. A convergence rate for the resulting controls and closed-loop systems isdemonstrated.

Mots clés: Stabilization; 2-D Navier-Stokes equations; Feedback control; Taylor expansion; Value function

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Tobias Breiten, Karl Kunisch, Laurent Pfeiffer. Feedback Stabilization of the Two-Dimensional Navier–Stokes Equations by Value Function Approximation. Applied Mathematics and Optimization, 2019, 80 (3), pp.599-641. ⟨10.1007/s00245-019-09586-x⟩. ⟨hal-02372083⟩