Authors

Fernando E. Serrano, Vicenc Puig, Jesús Muñoz Pacheco

Abstract

This paper proposes a robust dissipative boundary control for a system described by a 2-D Navier–Stokes equation with polytopic uncertainties under the port-Hamiltonian formulation. The goal of the controller is to obtain a fluid behavior that suppresses the vorticity of the waves. A complete robustness analysis, including both robust stability and robust performance, is presented. Additionally, a topological analysis of the polytopic uncertainties is performed to elucidate the open-loop system behavior. Meanwhile, the dissipative boundary-based controller is obtained by selecting an appropriate storage function. As a case study application, a wave tank is used to illustrate the approach proposed in the paper.

Keywords

2-d navier–stokes equation, polytopic uncertainties, robust control, wave tank

Citation

  • Journal: IFAC Journal of Systems and Control
  • Year: 2026
  • Volume: 37
  • Issue:
  • Pages: 100441
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.ifacsc.2026.100441

BibTeX

@article{Serrano_2026,
  title={{Dissipative boundary control of a 2-D Navier Stokes equation with polytopic uncertainties in the form of port-Hamiltonian formulation}},
  volume={37},
  ISSN={2468-6018},
  DOI={10.1016/j.ifacsc.2026.100441},
  journal={IFAC Journal of Systems and Control},
  publisher={Elsevier BV},
  author={Serrano, Fernando E. and Puig, Vicenc and Pacheco, Jesús Muñoz},
  year={2026},
  pages={100441}
}

Download the bib file

References

  • Ahmed-Ali T, Giri F, Krstic M, Lamnabhi-Lagarrigue F (2016) Adaptive Observer for a Class of Output-Delayed Systems with Parameter Uncertainty - A PDE Based Approach. IFAC-PapersOnLine 49(13):158–163. https://doi.org/10.1016/j.ifacol.2016.07.94 – 10.1016/j.ifacol.2016.07.944
  • Bastin G, Coron J-M, Hayat A (2025) The usefulness of viscosity for the robustness of boundary feedback control of an unstable fluid flow system. Automatica 173:112048. https://doi.org/10.1016/j.automatica.2024.11204 – 10.1016/j.automatica.2024.112048
  • Benabdelhadi A, Giri F, Ahmed-Ali T, Krstic M, El Fadil H, Chaoui F-Z (2021) Adaptive observer design for wave PDEs with nonlinear dynamics and parameter uncertainty. Automatica 123:109295. https://doi.org/10.1016/j.automatica.2020.10929 – 10.1016/j.automatica.2020.109295
  • Bhowmick P, Lanzon A (2021) Applying negative imaginary systems theory to non-square systems with polytopic uncertainty. Automatica 128:109570. https://doi.org/10.1016/j.automatica.2021.10957 – 10.1016/j.automatica.2021.109570
  • Boshenyatov B, Zhiltsov K (2019) Vortex suppression of tsunami-like waves by underwater barriers. Ocean Engineering 183:398–408. https://doi.org/10.1016/j.oceaneng.2019.05.01 – 10.1016/j.oceaneng.2019.05.011
  • Breit D (2025) Partial boundary regularity for the Navier–Stokes equations in irregular domains. Journal of Functional Analysis 289(12):111188. https://doi.org/10.1016/j.jfa.2025.11118 – 10.1016/j.jfa.2025.111188
  • Chang T, Kang K (2026) Singular velocity of the Stokes and Navier–Stokes equations near boundary in the half-space. Nonlinear Analysis 262:113939. https://doi.org/10.1016/j.na.2025.11393 – 10.1016/j.na.2025.113939
  • Chang X-H, Park JH, Zhou J (2015) Robust static output feedback H∞ control design for linear systems with polytopic uncertainties. Systems & Control Letters 85:23–32. https://doi.org/10.1016/j.sysconle.2015.08.00 – 10.1016/j.sysconle.2015.08.007
  • Chen H, Li Z, Zhang P (2026) Global axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations. Journal de Mathématiques Pures et Appliquées 205:103807. https://doi.org/10.1016/j.matpur.2025.10380 – 10.1016/j.matpur.2025.103807
  • Chen S, Preciado VM, Morari M, Matni N (2024) Robust model predictive control with polytopic model uncertainty through System Level Synthesis. Automatica 162:111431. https://doi.org/10.1016/j.automatica.2023.11143 – 10.1016/j.automatica.2023.111431
  • Du, Hydrodynamic performance and wave dynamics in the moonpool of a ship. part II: Geometry effects on resistance reduction and wave suppression. Ocean Engineering (2025)
  • – 10.1016/j.camwa.2022.11.026
  • Gutiérrez-Oribio D, Orlov Y, Stefanou I, Plestan F (2023) Robust boundary tracking control of wave PDE: Insight on forcing slow-aseismic response. Systems & Control Letters 178:105571. https://doi.org/10.1016/j.sysconle.2023.10557 – 10.1016/j.sysconle.2023.105571
  • Lin Y-X, Wang Y-G (2026) Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations. Journal of Mathematical Analysis and Applications 555(1):130042. https://doi.org/10.1016/j.jmaa.2025.13004 – 10.1016/j.jmaa.2025.130042
  • Macchelli A, Califano F (2018) Dissipativity-based boundary control of linear distributed port-Hamiltonian systems. Automatica 95:54–62. https://doi.org/10.1016/j.automatica.2018.05.02 – 10.1016/j.automatica.2018.05.029
  • Mao H, Teng J, Lin J, Yang S, Wu G, Wang W, Men Y (2026) Numerical study on the influence of mangrove density and belt width on wave attenuation characteristics under regular waves. Ocean Engineering 360:126120. https://doi.org/10.1016/j.oceaneng.2026.12612 – 10.1016/j.oceaneng.2026.126120
  • Panahi M, Porta GM, Riva M, Guadagnini A (2025) Modeling parametric uncertainty in PDEs models via Physics-Informed Neural Networks. Advances in Water Resources 195:104870. https://doi.org/10.1016/j.advwatres.2024.10487 – 10.1016/j.advwatres.2024.104870
  • Pozrikidis, (2001)
  • Raddanipour A, Shafiee M (2025) Robust iterative learning control of continuous delayed singular multi-agent systems with polytopic uncertainty and non-identical initial conditions: LMI approach. European Journal of Control 85:101354. https://doi.org/10.1016/j.ejcon.2025.10135 – 10.1016/j.ejcon.2025.101354
  • Serrano FE, Cayuela VP, Munoz-Pacheco JM (2026) Robust passivity-based boundary control of the 2-D Navier-Stokes equation with chaotic vortex. MATH 11(1):1777–1806. https://doi.org/10.3934/math.202607 – 10.3934/math.2026074
  • Wang Z, Zhu N (2025) Global well-posedness for 3D hyperbolic Navier-Stokes equations with ill-prepared initial data. Journal of Differential Equations 447:113647. https://doi.org/10.1016/j.jde.2025.11364 – 10.1016/j.jde.2025.113647
  • Xiao Y, Xu X, yuan Y (2025) Concurrent learning adaptive stabilization of 2 × 2 linear hyperbolic PDE systems with boundary parameter uncertainty. IFAC-PapersOnLine 59(8):196–201. https://doi.org/10.1016/j.ifacol.2025.08.09 – 10.1016/j.ifacol.2025.08.090
  • Yue L, Li YP (2026) On the buoyancy production term for Reynolds-averaged modelling of breaking waves. Coastal Engineering 205:104935. https://doi.org/10.1016/j.coastaleng.2025.10493 – 10.1016/j.coastaleng.2025.104935
  • Zattoni E, Bartolucci V, Perdon AM, Conte G, Scaradozzi D (2025) Model Matching Problems for Max-plus Linear Systems with Polytopic Uncertainty in the Dynamics. IFAC-PapersOnLine 59(12):43–48. https://doi.org/10.1016/j.ifacol.2025.09.56 – 10.1016/j.ifacol.2025.09.565
  • Zattoni E, Otsuka N, Perdon AM, Conte G (2024) Model matching problems for impulsive linear systems with polytopic uncertainties. Nonlinear Analysis: Hybrid Systems 52:101465. https://doi.org/10.1016/j.nahs.2024.10146 – 10.1016/j.nahs.2024.101465
  • Zhang X-Y, Li Y-X, Ahn CK (2023) Prescribed finite-time boundary control of constrained flexible satellite systems with translational motion modeling. Aerospace Science and Technology 140:108470. https://doi.org/10.1016/j.ast.2023.10847 – 10.1016/j.ast.2023.108470
  • Zhao R-X, Guo B-Z, Paunonen L (2025) Robust output regulation for multi-dimensional heat equation under boundary control. Automatica 171:111956. https://doi.org/10.1016/j.automatica.2024.11195 – 10.1016/j.automatica.2024.111956
  • Zhou W-J, Wu K-N, Liu X-Z (2025) Robust sliding mode boundary control for uncertain reaction–diffusion systems. Journal of the Franklin Institute 362(6):107623. https://doi.org/10.1016/j.jfranklin.2025.10762 – 10.1016/j.jfranklin.2025.107623