# Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Function Spaces

## 3. BBC Design Methodology for PDEs

#### 3.1. Stable Objective System

**Lemma 1**

**Poincaré Inequality**[43]). For any $w\in {\mathcal{H}}^{1}(0,1)$ (Sobolev space) the following relations hold

#### 3.2. Backstepping Transformation

#### 3.3. Integral Equation Solution

#### 3.4. Neumann Controller

## 4. Modified Burgers’ Equation

## 5. Identifier PDE

## 6. Estimation Error PDE

## 7. Adaptive Control Laws

## 8. Simulation Results

## 9. Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ADI | Alternating Direction Implicit |

BBC | Backstepping Boundary Control |

BC | Boundary Condition |

DPSs | Distributed Parameters Systems |

MRAC | Model Reference Adaptive Control |

ODEs | Ordinary Differential Equations |

OSC | Orthogonal Spline Collocation |

PIDE | Partial Integro-Differential Equations |

PDEs | Partial Differential Equations |

RAD | Reaction-Advection-Diffusion |

RD | Reaction-Diffusion |

WSK | Weakly Singular Kernel |

## References

- Evans, L.C. Partial Differential Equations, 2nd ed.; American Mathematical Society: Providence, RI, USA, 2010. [Google Scholar]
- Koga, S.; Krstić, M. Materials Phase Change PDE Control & Estimation: From Additive Manufacturing to Polar Ice; Springer Birkhäuser: Cham, Switzerland, 2020. [Google Scholar]
- Yu, H.; Krstić, M. Traffic Congestion Control by PDE Backstepping; Springer Birkhäuser: Cham, Switzerland, 2022. [Google Scholar]
- Anfinsen, H.; Aamo, O.M. Adaptive Control of Hyperbolic PDEs; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Paunonen, L.; Meurer, T. IEEE Control Systems Society Technical Committee on Distributed Parameter Systems. IEEE Control Syst.
**2023**, 43, 18–20. [Google Scholar] [CrossRef] - Della Santina, C.; Duriez, C.; Rus, D. Model-Based Control of Soft Robots. IEEE Control Syst.
**2023**, 43, 30–65. [Google Scholar] [CrossRef] - Strauss, W.A. Partial Differential Equations: An Introduction, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Byrnes, C.I.; Gilliam, D.S.; Shubov, V.I. Boundary control for a viscous Burgers’ equation. In Identification Control for Systems Governed by Partial Differential Equations; Banks, H.T., Fabiano, R.H., Ito, K., Eds.; SIAM: Philadelphia, PA, USA, 1993. [Google Scholar]
- Guenther, R.B.; Lee, J.W. Partial Differential Equations of Mathematical Physics and Integral Equations; Dover: Mineola, NY, USA, 1996. [Google Scholar]
- Krstić, M. On Global Stabilization of Burgers’ Equation by Boundary Control. In Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, USA, 18 December 1998; pp. 3498–3499. [Google Scholar]
- Gustafson, K.E. Introduction to Partial Differential Equations and Hilbert Space Methods, 3rd ed.; Dover: Mineola, NY, USA, 1999. [Google Scholar]
- Polyanin, A.D.; Zaitsev, V.F. Handbook of Nonlinear Partial Differential Equations, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2012. [Google Scholar]
- Humi, M. Introduction to Mathematical Modeling; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Tian, Q.; Yang, X.; Zhang, H.; Xu, D. An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties. Comput. Appl. Math.
**2023**, 42, 246. [Google Scholar] [CrossRef] - Yang, X.; Wu, L.; Zhang, H. A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput.
**2023**, 457, 651–674. [Google Scholar] [CrossRef] - Zhang, H.; Liu, Y.; Yang, X. An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space. J. Appl. Math. Comput.
**2023**, 69, 651–674. [Google Scholar] [CrossRef] - Zhang, H.; Yang, X.; Tang, Q.; Xu, D. A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation. Comput. Math. Appl.
**2022**, 109, 180–190. [Google Scholar] [CrossRef] - Krstić, M.; Smyshlyaev, A. Boundary Control for PDEs: A Course on Backstepping Designs; SIAM: Philadelphia, PA, USA, 2008. [Google Scholar]
- Krstić, M.; Smyshlyaev, A. Boundary Control of PDEs: The Backstepping Approach. In The Control Handbook, 2nd ed.; Levine, W.S., Ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Krstić, M.; Magnis, L.; Vazquez, R. Nonlinear Control of the Burgers PDE-Part I: Full-State Stabilization. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 285–290. [Google Scholar]
- Krstić, M.; Magnis, L.; Vazquez, R. Nonlinear Stabilization of Shock-Like Unstable Equilibria in the Viscous Burgers PDE. IEEE Trans. Autom. Control
**2008**, 53, 1678–1683. [Google Scholar] [CrossRef] - Krstić, M.; Magnis, L.; Vazquez, R. Nonlinear Control of the Burgers PDE-Part II: Observer Design, Trajectory Generation, and Tracking. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 3076–3081. [Google Scholar]
- Kobayashi, T. Adaptive regulator design of a viscous Burgers’ system by boundary control. IMA J. Math. Control. Inf.
**2001**, 18, 427–437. [Google Scholar] [CrossRef] - Morris, K. Control of Systems Governed by Partial Differential Equations. In The Control Handbook, 2nd ed.; Levine, W.S., Ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Aulisa, E.; Gilliam, D. A Practical Guide to Geometric Regulation for Distributed Parameter Systems; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Hong, K.S.; Bentsman, J. Direct Adaptive Control of Parabolic Systems: Algorithm Synthesis and Convergence and Stability Analysis. IEEE Trans. Autom. Control
**1994**, 39, 2018–2033. [Google Scholar] [CrossRef] - Orlov, Y.V. Model Reference Adaptive Control of Distributed Parameter Systems. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA, 12 December 1997; pp. 263–268. [Google Scholar]
- Böhm, M.; Demetriou, M.A.; Reich, S.; Rosen, I.G. Model Reference Adaptive Control of Distributed Parameter Systems. SIAM J. Control Optim.
**1998**, 36, 33–81. [Google Scholar] [CrossRef] - Liu, W.J.; Krstić, M. Adaptive control of Burgers’ equation with unknown viscosity. Int. J. Adapt. Control Signal Process.
**2001**, 15, 745–766. [Google Scholar] [CrossRef] - Krstić, M. Lyapunov Adaptive Stabilization of Parabolic PDEs-Part I: A Benchmark for Boundary Control. In Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 12–15 December 2005; pp. 3164–3169. [Google Scholar]
- Krstić, M. Lyapunov Adaptive Stabilization of Parabolic PDEs-Part II: Output Feedback and Other Benchmark Problems. In Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 12–15 December 2005; pp. 3170–3175. [Google Scholar]
- Smyshlyaev, A.; Krstić, M. Output–Feedback Adaptive Control for Parabolic PDEs with Spatially Varying Coefficients. In Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 13–15 December 2006; pp. 3099–3104. [Google Scholar]
- Smyshlyaev, A.; Krstić, M. Lyapunov Adaptive Boundary Control for Parabolic PDEs with Spatially Varying Coefficients. In Proceedings of the American Control Conference, Minneapolis, MN, USA, 14–16 June 2006; pp. 41–48. [Google Scholar]
- Smyshlyaev, A.; Krstić, M. Adaptive boundary control for unstable parabolic PDEs-Part II: Estimation-based designs. Automatica
**2007**, 43, 1543–1556. [Google Scholar] [CrossRef] - Smyshlyaev, A.; Krstić, M. Adaptive boundary control for unstable parabolic PDEs-Part III: Output feedback examples with swapping identifiers. Automatica
**2007**, 43, 1557–1564. [Google Scholar] [CrossRef] - Krstić, M.; Smyshlyaev, A. Adaptive Boundary Control for Unstable Parabolic PDEs—Part I: Lyapunov Design. IEEE Trans. Autom. Control
**2008**, 53, 1575–1591. [Google Scholar] [CrossRef] - Morris, K.A. Controller Design for Distributed Parameter Systems; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Krstic, M. On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett.
**1999**, 37, 123–142. [Google Scholar] [CrossRef] - Balogh, A.; Krstić, M. Burgers’ equation with nonlinear boundary feedback: H
^{1}stability, well posedness, and simulation. Math. Probl. Eng.**2000**, 6, 189–200. [Google Scholar] [CrossRef] - Liu, W.J.; Krstić, M. Backstepping boundary control of Burgers’ equation with actuator dynamics. Syst. Control Lett.
**2000**, 41, 291–303. [Google Scholar] [CrossRef] - Temam, R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics; Springer: New York, NY, USA, 1998. [Google Scholar]
- Cruz-Quintero, E.; Jurado, F. Boundary Control for a Certain Class of Reaction-Advection-Diffusion System. Mathematics
**2020**, 8, 1854. [Google Scholar] [CrossRef] - Smyshlyaev, A.; Krstić, M. Adaptive Control of Parabolic PDEs; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables; Dover Publications: Mineola, NY, USA, 1965. [Google Scholar]
- Zachmanoglou, E.C.; Thoe, D.W. Introduction to Partial Differential Equations with Applications; Dover: Mineola, NY, USA, 1986. [Google Scholar]
- Arfken, G.B.; Weber, H.J.; Harris, F.E. Mathematical Methods for Physicists: A Comprehensive Guide, 7th ed.; Academic Press: Waltham, MA, USA, 2013. [Google Scholar]
- Ioannou, P.A.; Sun, J. Robust Adaptive Control; Prentice Hall: Englewood Cliffs, NJ, USA, 1995. [Google Scholar]
- Murillo-García, O.F.; Jurado, F. Adaptive Boundary Control for a Certain Class of Reaction-Advection-Diffusion System. Mathematics
**2021**, 9, 2224. [Google Scholar] [CrossRef]

**Figure 1.**Adaptive BBC scheme for the modified Burgers’ system (124)–(126).

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**MDPI and ACS Style**

Jurado, F.; Murillo-García, O.F.
Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation. *Fractal Fract.* **2023**, *7*, 834.
https://doi.org/10.3390/fractalfract7120834

**AMA Style**

Jurado F, Murillo-García OF.
Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation. *Fractal and Fractional*. 2023; 7(12):834.
https://doi.org/10.3390/fractalfract7120834

**Chicago/Turabian Style**

Jurado, Francisco, and Oscar F. Murillo-García.
2023. "Adaptive Backstepping Boundary Control for a Class of Modified Burgers’ Equation" *Fractal and Fractional* 7, no. 12: 834.
https://doi.org/10.3390/fractalfract7120834