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arxiv: 2604.15846 · v1 · submitted 2026-04-17 · 🧮 math.OC · cs.NA· math.NA

Finite-Dimensional MOR-Based RHC for Steering 2D Navier-Stokes Equations to Desired Trajectories

Behzad Azmi , Stefan Frei , Felix Sauer This is my paper

Pith reviewed 2026-05-10 08:20 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords 2D Navier-Stokes equationsreceding horizon controlmodel order reductionproper orthogonal decompositiontrajectory trackingfinite-dimensional controllocal exponential stabilization
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The pith

A finite collection of fixed indicator-function actuators combined with receding horizon control locally exponentially stabilizes 2D Navier-Stokes flows to any reference trajectory.

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

The paper shows that receding horizon control can steer two-dimensional incompressible viscous flows along prescribed time-dependent paths when the control input is restricted to linear combinations of a few fixed indicator functions supported on a given subdomain. It proves that the resulting closed-loop system is locally exponentially stable around the target trajectory and that the finite-horizon costs remain within a fixed factor of the optimal infinite-horizon cost. To address the high computational burden of the full-order model, the authors replace the state equation with a proper-orthogonal-decomposition reduced model and verify in numerical tests that the reduced controller still achieves comparable stabilization performance at substantially lower cost.

Core claim

A receding-horizon feedback law obtained by repeatedly solving a finite-horizon optimal control problem over a moving time window, with the control constrained to a finite-dimensional space spanned by indicator functions on a prescribed subdomain, renders the two-dimensional Navier-Stokes equations locally exponentially stable about any sufficiently regular reference trajectory; the same scheme satisfies a suboptimality estimate relating its cost to that of the corresponding infinite-horizon problem.

What carries the argument

The receding-horizon feedback operator computed from a finite-dimensional control space of indicator functions on a fixed control subdomain, optionally realized via a proper-orthogonal-decomposition reduced-order model of the state equation.

If this is right

  • Initial data sufficiently close to the reference trajectory converge exponentially fast to it under the closed-loop dynamics.
  • The realized cost of the receding-horizon scheme remains bounded by a constant multiple of the optimal infinite-horizon cost.
  • A proper-orthogonal-decomposition surrogate preserves the observed stabilization behavior while lowering the dimension of the optimization problem solved at each step.
  • The method applies to two flow configurations of moderate complexity, as confirmed by the reported numerical experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • An offline optimization of actuator locations could be added as a preprocessing stage to enlarge the basin of attraction or to improve robustness when the original placement is only marginally sufficient.
  • Replacing proper orthogonal decomposition with other reduction techniques such as balanced truncation or snapshot-based data-driven bases could be tested to see whether the same suboptimality bound continues to hold in practice.
  • The fixed-subdomain assumption suggests that extensions to time-varying actuator supports or to domains containing moving obstacles would require new controllability arguments beyond those used here.

Load-bearing premise

The chosen finite set of indicator-function actuators must be sufficient to achieve local exponential stabilization around the reference trajectory.

What would settle it

A specific reference trajectory together with a small initial perturbation for which every admissible finite combination of actuator coefficients causes the solution to diverge from the trajectory would disprove the local stabilizability claim.

Figures

Figures reproduced from arXiv: 2604.15846 by Behzad Azmi, Felix Sauer, Stefan Frei.

Figure 1
Figure 1. Figure 1: Left: Different actuator layouts used in Example 1. Right: Evolution of ∥v(t)∥H for the two layouts and T = 1. T J o T∞ (urh; 0, y0) 0.25 0.276436303216 1.0 0.194958441937 2.0 0.194547920119 no controls 0.559781470382 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Evolution of ∥vrh(t)∥H for RHC with different prediction horizons. Right: Correspond￾ing values of the cost function uncontrolled flow. Moreover, the configuration with 48 actuator supports achieves a noticeably faster rate of stabilization than the configuration with 25 actuator supports. Prediction horizon We next fix the square actuator layout and compare the performance of the RHC scheme for the … view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of ∥v(t)∥H for different viscosities ν = 10−1 , 10−2 and 10−3 (left to right) in Example 1. relatively quickly. In this regime, the improvement obtained by RHC over the uncontrolled dynamics is therefore expected to be moderate. For smaller viscosities, by contrast, the dynamics are significantly less dissipative, and the uncontrolled flow requires substantially more time to approach the stationa… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Evolution of ∥v(t)∥H for the FOM and MOR-based RHC with different POD-basis sizes applied to Example 1. Right: Runtime (in s) of the FOM algorithm 2 and the MOR-based algorithm 3 with l = 20 for Example 1 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left: Non-stationary flow y˜ and initial state y0. Right: Stationary flow yˆ used in Example 2. outflow condition: y(0, x2) =  4ymaxx2(0.41 − x2) 0.412 , 0  on Γin, ν ∂y ∂n − pn = 0 on Γout. (33) We fix the viscosity at ν = 1 750 , corresponding to a Reynolds number of Re = 75. For this Reynolds number, the flow develops an unsteady regime with time-periodic vortex shedding behind the obstacle; see [PIT… view at source ↗
Figure 6
Figure 6. Figure 6: Spatial discretization and actuator layout for the channel flow (Example 2) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of ∥v(t)∥H for the uncontrolled state and the MOR-based RHC algorithm with T = 0.3, ℓ = 50, provided the ROM basis is only computed once after the first interval (0, T). trajectory then deteriorates and slowly approaches the uncontrolled one. We also note that, on the first interval, the full-order RHC and MOR-based RHC solutions coincide by construction. In the remaining experiments, we therefor… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of ∥v(t)∥H for the FOM RHC and MOR-based RHC algorithms (FOM-ROM) applied to Example 2 with T ∈ {0.3, 0.6}, where the ROM basis is updated regularly. Method Total First horizon POD updates/assembly Other horizons FOM 4 654 441 – 4 213 FOM-ROM 735 441 290 4 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

This paper investigates the local exponential stabilization of the two-dimensional Navier--Stokes equations to a given reference trajectory by means of receding horizon control (RHC). The control is realized as a linear combination of finitely many actuators, represented by indicator functions supported on subsets of a prescribed control subdomain. We establish local exponential stabilizability and suboptimality for the resulting RHC scheme. Numerical experiments for two flow configurations of increasing complexity illustrate the theoretical findings and assess the practical performance of the method. In addition, we propose a model-order-reduced RHC approach based on proper orthogonal decomposition, which significantly reduces the computational cost while maintaining favorable closed-loop stabilization performance in the numerical experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a receding-horizon control (RHC) scheme for steering the 2D Navier-Stokes equations to prescribed trajectories using a finite number of indicator-function actuators supported on a prescribed control subdomain. It claims to establish local exponential stabilizability and suboptimality of the resulting closed-loop system, proposes a proper orthogonal decomposition (POD) model-order reduction (MOR) to lower computational cost, and illustrates the approach with numerical experiments on two flow configurations of increasing complexity.

Significance. If the stabilizability result holds, the work supplies a computationally feasible route to trajectory tracking for incompressible flows by combining RHC with finite-dimensional actuation and POD-based reduction. The numerical tests on flows of graded complexity constitute a concrete strength, showing that the reduced-order controller retains stabilization performance. The absence of a general proof or verifiable condition for actuator sufficiency, however, confines the theoretical contribution to the specific cases treated numerically.

major comments (2)
  1. [§3] §3 (theoretical analysis of local exponential stabilizability): the central claim that the finite-dimensional control operator (linear combination of indicator functions) renders the linearized Navier-Stokes operator exponentially stable around an arbitrary reference trajectory is asserted without a general controllability theorem, spectral gap condition, or explicit verification that the chosen actuators reach all unstable modes of the trajectory-dependent linearization. This assumption is load-bearing for both the stabilizability statement and the subsequent suboptimality estimates.
  2. [§4] §4 (suboptimality analysis): the suboptimality bound for the RHC scheme is stated in the abstract and introduction but is not accompanied by an explicit error estimate or remainder term that accounts for the finite-actuator approximation and the linearization error; without this, the quantitative performance claim cannot be verified independently of the numerical examples.
minor comments (3)
  1. [§2] The notation for the control subdomain Ω_c and the indicator functions χ_i is introduced without an accompanying diagram or explicit formula in the problem-statement section, which would improve readability.
  2. [§5] Figure captions for the two flow configurations omit the precise values of the Reynolds number, time horizon, and POD truncation rank used in each experiment.
  3. [References] A few standard references on finite-dimensional controllability of the Navier-Stokes linearization are absent from the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment point by point below, clarifying the scope of our theoretical results and indicating revisions where appropriate to improve the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (theoretical analysis of local exponential stabilizability): the central claim that the finite-dimensional control operator (linear combination of indicator functions) renders the linearized Navier-Stokes operator exponentially stable around an arbitrary reference trajectory is asserted without a general controllability theorem, spectral gap condition, or explicit verification that the chosen actuators reach all unstable modes of the trajectory-dependent linearization. This assumption is load-bearing for both the stabilizability statement and the subsequent suboptimality estimates.

    Authors: Our stabilizability result is established specifically for the reference trajectories and actuator configurations analyzed in the paper, rather than for arbitrary trajectories. The proof of local exponential stability for the closed-loop RHC system relies on the fact that the finite-dimensional actuators (indicator functions) can control the unstable modes of the trajectory-dependent linearization in the cases considered; this is verified through the structure of the Stokes operator and numerical spectral analysis for the chosen setups. We do not claim or provide a general controllability theorem because the unstable modes vary with the trajectory, and deriving a universal condition would require additional assumptions on actuator placement that lie outside the scope of this work, which focuses on a computationally feasible RHC approach for concrete 2D flow problems. In the revised manuscript, we have added a clarifying remark in §3 explicitly stating the trajectory-specific nature of the result and outlining how actuator sufficiency is confirmed for the examples. revision: partial

  2. Referee: [§4] §4 (suboptimality analysis): the suboptimality bound for the RHC scheme is stated in the abstract and introduction but is not accompanied by an explicit error estimate or remainder term that accounts for the finite-actuator approximation and the linearization error; without this, the quantitative performance claim cannot be verified independently of the numerical examples.

    Authors: The suboptimality bound follows from standard arguments in receding-horizon control theory once local exponential stability of the closed-loop system is established; the finite-actuator and linearization effects are implicitly controlled by the stability margin and the choice of prediction horizon. We agree that an explicit decomposition of the error terms would make the quantitative claims easier to verify. In the revision, we will augment the suboptimality analysis in §4 with a remark that isolates the contributions of the finite-dimensional actuation and linearization error to the performance bound, referencing the stability constants derived earlier. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external infinite-dimensional control theory

full rationale

The paper explicitly assumes that the finite indicator-function actuators on the prescribed subdomain suffice for local exponential stabilization of the linearized 2D Navier-Stokes operator around the reference trajectory. From this hypothesis, standard receding-horizon control arguments (local stabilizability implying suboptimality) are applied to obtain the RHC results. No derivation step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the model-order reduction via POD is a standard, independent technique. The derivation chain is therefore self-contained against external benchmarks in infinite-dimensional control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard well-posedness results for the 2D Navier-Stokes system and on the existence of a reference trajectory; no new free parameters or postulated entities are introduced.

axioms (1)
  • standard math The 2D Navier-Stokes equations admit unique mild or weak solutions for given initial data, forcing, and boundary conditions in appropriate Sobolev spaces.
    This background result is required for the controlled system to be well-defined before any stabilization analysis begins.

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