Approximation of Feedback Gains Using Spectral Projections: Application to the Oseen System
Year: 2024
Type: article
Abstract: . We consider a parabolic controlled system (called [Formula: see text]) in a Hilbert space [Formula: see text] and a family of approximate parabolic controlled systems (called [Formula: see text]) in a Hilbert space [Formula: see text]. We are in the case of nonconforming approximations, that is, when [Formula: see text]. We assume that both [Formula: see text] and [Formula: see text] are Hilbert subspaces of another Hilbert space [Formula: see text]. We assume that [Formula: see text] and [Formula: see text] satisfy approximation errors, [Formula: see text] is exponentially stabilizable, and [Formula: see text] is uniformly parabolic. We define a reduced order model, called [Formula: see text] (resp., [Formula: see text]), based on spectral projections for [Formula: see text] (resp., [Formula: see text]). We construct a Riccati based feedback [Formula: see text] for [Formula: see text] which exponentially stabilizes both [Formula: see text] and [Formula: see text], and we construct an associated Riccati based feedback [Formula: see text] for [Formula: see text] which exponentially stabilizes both [Formula: see text] and [Formula: see text]. We prove convergence rates for [Formula: see text], and we prove that, for all [Formula: see text], where [Formula: see text] is small enough, [Formula: see text] also exponentially stabilizes [Formula: see text]. We apply these results to the boundary feedback stabilization of the Oseen system (the linearized Navier–Stokes system around an unstable stationary solution) and to its numerical approximation by a finite element method (F.E.M.). In that case [Formula: see text], where [Formula: see text] is the meshsize of the F.E.M.
DOI: 10.1137/23m1609695
Authors: Mehdi Badra, Jean-Pierre Raymond
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404048348