Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

Francisco Fuica; Nicolai Jork

arxiv: 2605.01633 · v2 · pith:AWG5P5HGnew · submitted 2026-05-02 · 🧮 math.NA · cs.NA· math.OC

Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

Francisco Fuica , Nicolai Jork This is my paper

Pith reviewed 2026-05-08 19:35 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC

keywords optimalerrorcontroldiscretizationequationsproblemproveconsider

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Error estimates are proven for variational discretization of an unregularized optimal control problem for the stationary Navier-Stokes equations, for nonsingular locally optimal controls satisfying a growth condition that implies bang-bang structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Fluid flows are often modeled by the Navier-Stokes equations, and engineers sometimes want to find the best way to control them, such as by choosing forces or boundary conditions. This paper studies the case without extra smoothing terms that usually make the math easier. It shows that optimal controls exist, gives mathematical conditions to check if a solution is the best one, and then looks at how to approximate those solutions on a computer using a specific discretization method. Under assumptions that the optimal control is nonsingular and has a certain growth property leading to bang-bang behavior, the paper gives bounds on how much the numerical approximation differs from the true optimal control.

Core claim

We prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure.

Load-bearing premise

The locally optimal controls are nonsingular and satisfy a growth condition implying bang-bang structure (as stated in the abstract for the error estimates to hold).

read the original abstract

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure. We also propose a residual-type a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove suitable reliability properties for such an error estimator.

Editorial analysis

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Desk Editor's Note private letter to a colleague

The paper derives conditional a priori error estimates for variational discretization of an unregularized steady Navier-Stokes control problem under nonsingularity and bang-bang growth assumptions.

read the letter

I read the paper on error estimates for the unregularized optimal control problem with stationary Navier-Stokes. The central new result is the a priori error bounds for the variational discretization when the locally optimal control is nonsingular and satisfies the growth condition that forces bang-bang structure. They first establish existence of solutions along with first- and second-order necessary and sufficient optimality conditions without regularization. That part sets up the discretization analysis in a straightforward way. Convergence of the discrete controls follows, and the error estimates are then obtained by using the growth condition to control the difference between continuous and discrete solutions. This extends earlier results on regularized problems or other equations to the unregularized Navier-Stokes setting under those assumptions. The limitation is that the rates hold only when the control meets the stated conditions. Without them the paper makes no claim about rates, which is the usual situation for unregularized bang-bang problems. The work stays strictly in the steady case and uses standard PDE arguments with no machine-checked proofs or external data. This is for researchers in numerical analysis of PDE-constrained optimization who work on fluid control problems. A reader who needs theoretical error bounds for discretizations with bang-bang controls would get something usable here. The claims are scoped clearly and the derivations appear careful, so it deserves peer review.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard existence and regularity results for the stationary Navier-Stokes equations and functional-analytic tools for optimal control; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Existence of solutions to the stationary Navier-Stokes equations under standard assumptions on data and domain.
Invoked implicitly for the optimal control problem to be well-posed.
standard math Standard first- and second-order optimality conditions for PDE-constrained optimization.
Used to derive the necessary and sufficient conditions.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost (Jcost) and Foundation/AlphaCoordinateFixation no parallel; J-cost / α-pin machinery is absent here unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a priori error estimates for locally optimal solutions under relatively mild assumptions, including bounds in the maximum norm for the adjoint velocity field and in the L^1(Ω)-norm for the control variable.
Foundation/BranchSelection, Cost/FunctionalEquation exponent γ here is a subregularity index, unrelated to RS bilinear-branch α or φ-exponents unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

growth condition J'(¯u)(u−¯u) + J''(¯u)(u−¯u)^2 ≥ c‖u−¯u‖^{1+1/γ}_{L^1}, with γ∈(n/(n+2),1]

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