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May  2022, 15(5): 1085-1102. doi: 10.3934/dcdss.2021092

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

1. 

Department of Mathematics, Higher Institute of computer scince and multimedia of Sfax, University of Sfax, Sfax, Tunisia

2. 

Department of Mathematics, College of Sciences, Qassim University, Buraidah, Saudi Arabia

* Corresponding author: abdallah.benabdallah@ipeis.rnu.tn

Received  March 2021 Revised  June 2021 Published  May 2022 Early access  August 2021

In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and $ 1-d $ unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.

Citation: Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1085-1102. doi: 10.3934/dcdss.2021092
References:
[1]

T. Ahmed-AliF. GiriM. Krstic and F. Lamnabhi-Lagarrigue, Observer design for a class of nonlinear ODE-PDE cascade systems, Systems & Control Letters, 83 (2015), 19-27.  doi: 10.1016/j.sysconle.2015.06.003.

[2]

N. Bekiaris-Liberis and M. Krstic, Compensation of wave actuator dynamics for nonlinear systems, IEEE Transaction on Automatic Control, 59 (2014), 1555-1570.  doi: 10.1109/TAC.2014.2309057.

[3]

A. Benabdallah, Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller, International Journal of Robust and Nonlinear Control, 30 (2020), 3023-3038.  doi: 10.1002/rnc.4901.

[4]

X. Cai and M. Krstic, Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary, Automatica, 68 (2016), 27-38.  doi: 10.1016/j.automatica.2016.01.043.

[5]

X. CaiL. LiaoJ. Zhang and W. Zhang, Observer design for a class of nonlinear system in cascade with counter-convecting transport dynamics, Kybernetika, 52 (2016), 76-88.  doi: 10.14736/kyb-2016-1-0076.

[6]

C. ChalonsM.-L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Interfaces and Free Boundaries, 19 (2017), 553-570.  doi: 10.4171/IFB/392.

[7]

J. DaafouzM. Tucsnak and J. Valein, Nonlinear control of a coupled PDE-ODE system modeling a switched power converter with a transmission line, Systems & Control Letters, 70 (2014), 92-99.  doi: 10.1016/j.sysconle.2014.05.009.

[8]

M. DiagneN. Bekiaris-LiberisA. Otto and M. Krstic, Control of transport pde/nonlinear ODE cascades with state-dependent propagation speed, IEEE Trans. Automat. Control, 62 (2017), 6278-6293.  doi: 10.1109/TAC.2017.2702103.

[9]

A. HasanaO. M. Aamoa and M. Krstic, Boundary observer design for hyperbolic PDE-ODE cascade systems, Automatica, 68 (2016), 75-86.  doi: 10.1016/j.automatica.2016.01.058.

[10]

F. Hassine, Rapid exponential stabilization of a 1-d transmission wave equation with in-domain anti-damping, Asian Journal of Control, 19 (2017), 2017-2027.  doi: 10.1002/asjc.1509.

[11]

M. Krstic, Compensating a string pde in the actuation or sensing path of an unstable ODE, IEEE Transaction on Automatic Control, 54 (2009), 1362-1368.  doi: 10.1109/TAC.2009.2015557.

[12]

M. Krstic, Compensating actuator and sensor dynamics governed by diffusion PDES, Systems & Control Letters, 58 (2009), 372-377.  doi: 10.1016/j.sysconle.2009.01.006.

[13]

H. Lei and W. Lin, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica, 42 (2006), 1783-1789.  doi: 10.1016/j.automatica.2006.05.006.

[14]

X. Liu and C. Xie, Control law in analytic expression of a system coupled by reaction-diffusion equation, Systems & Control Letters, 137 (2020), 104643, 5 pp. doi: 10.1016/j.sysconle.2020.104643.

[15]

F. MazencL. Praly and W. P. Dayawansa, Global stabilization by output feedback : Examples and counter-examples, Systems & Control Letters, 23 (1994), 119-125.  doi: 10.1016/0167-6911(94)90041-8.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

B. SaldivarS. Mondié and J. C. Avila Vilchis, The control of drilling vibrations: A coupled PDE-ODE modeling approach, International Journal Applied Mathematics Computer Science, 26 (2016), 335-349.  doi: 10.1515/amcs-2016-0024.

[18]

A. Smyshlyaev and M. Krstic, Closed-form boundary state feedbacks for a class of $1-d$ partial integro-differential equations, IEEE Transaction on Automatic Control, 49 (2004), 2185-2202.  doi: 10.1109/TAC.2004.838495.

[19]

G. A. Susto and M. Krstic, Control of PDE-ODE cascades with neumann interconnections, Journal of Franklin Institute, 347 (2010), 284-314.  doi: 10.1016/j.jfranklin.2009.09.005.

[20]

S. Tang and C. Xie, Stabilization for a coupled PDE-ODE control system, Journal of the Franklin Institute, 348 (2011), 2142-2155.  doi: 10.1016/j.jfranklin.2011.06.008.

[21]

S. Tang and C. Xie, State and output feedback boundary control for a coupled PDE-ODE system, Systems & Control Letters, 60 (2011), 540-545.  doi: 10.1016/j.sysconle.2011.04.011.

[22]

Y. Tang and G. Mazanti, Stability analysis of coupled linear ODE-hyperbolic PDE systems with two time scales, Automatica, 85 (2017), 386-396.  doi: 10.1016/j.automatica.2017.07.052.

[23]

G. Weiss and X. Zhao, Well-posedness and controllability of a class of coupled linear systems, SIAM Journal on Control and Optimization, 48 (2009), 2719-2750.  doi: 10.1137/090752833.

[24]

H.-N. Wu and J.-W. Wang, Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion pde-governed sensor dynamics, Nonlinear Dynamics, 72 (2013), 615-628.  doi: 10.1007/s11071-012-0740-4.

[25]

A. Zhao and C. Xie, Stabilization of coupled linear plant and reaction-diffusion process, Journal of the Franklin Institute, 351 (2014), 857-877.  doi: 10.1016/j.jfranklin.2013.09.012.

[26]

X. Zhao and G. Weiss, Controllability and observability of a well-posed system coupled with a finite-dimensional system, IEEE Transactions on Automatic Control, 56 (2011), 88-99.  doi: 10.1109/TAC.2010.2051352.

[27]

Z. Zhoua and S. Tang, Boundary stabilization of a coupled wave-ode system with internal anti-damping, International Journal of Control, 85 (2012), 1683-1693.  doi: 10.1080/00207179.2012.696704.

show all references

References:
[1]

T. Ahmed-AliF. GiriM. Krstic and F. Lamnabhi-Lagarrigue, Observer design for a class of nonlinear ODE-PDE cascade systems, Systems & Control Letters, 83 (2015), 19-27.  doi: 10.1016/j.sysconle.2015.06.003.

[2]

N. Bekiaris-Liberis and M. Krstic, Compensation of wave actuator dynamics for nonlinear systems, IEEE Transaction on Automatic Control, 59 (2014), 1555-1570.  doi: 10.1109/TAC.2014.2309057.

[3]

A. Benabdallah, Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller, International Journal of Robust and Nonlinear Control, 30 (2020), 3023-3038.  doi: 10.1002/rnc.4901.

[4]

X. Cai and M. Krstic, Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary, Automatica, 68 (2016), 27-38.  doi: 10.1016/j.automatica.2016.01.043.

[5]

X. CaiL. LiaoJ. Zhang and W. Zhang, Observer design for a class of nonlinear system in cascade with counter-convecting transport dynamics, Kybernetika, 52 (2016), 76-88.  doi: 10.14736/kyb-2016-1-0076.

[6]

C. ChalonsM.-L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Interfaces and Free Boundaries, 19 (2017), 553-570.  doi: 10.4171/IFB/392.

[7]

J. DaafouzM. Tucsnak and J. Valein, Nonlinear control of a coupled PDE-ODE system modeling a switched power converter with a transmission line, Systems & Control Letters, 70 (2014), 92-99.  doi: 10.1016/j.sysconle.2014.05.009.

[8]

M. DiagneN. Bekiaris-LiberisA. Otto and M. Krstic, Control of transport pde/nonlinear ODE cascades with state-dependent propagation speed, IEEE Trans. Automat. Control, 62 (2017), 6278-6293.  doi: 10.1109/TAC.2017.2702103.

[9]

A. HasanaO. M. Aamoa and M. Krstic, Boundary observer design for hyperbolic PDE-ODE cascade systems, Automatica, 68 (2016), 75-86.  doi: 10.1016/j.automatica.2016.01.058.

[10]

F. Hassine, Rapid exponential stabilization of a 1-d transmission wave equation with in-domain anti-damping, Asian Journal of Control, 19 (2017), 2017-2027.  doi: 10.1002/asjc.1509.

[11]

M. Krstic, Compensating a string pde in the actuation or sensing path of an unstable ODE, IEEE Transaction on Automatic Control, 54 (2009), 1362-1368.  doi: 10.1109/TAC.2009.2015557.

[12]

M. Krstic, Compensating actuator and sensor dynamics governed by diffusion PDES, Systems & Control Letters, 58 (2009), 372-377.  doi: 10.1016/j.sysconle.2009.01.006.

[13]

H. Lei and W. Lin, Universal adaptive control of nonlinear systems with unknown growth rate by output feedback, Automatica, 42 (2006), 1783-1789.  doi: 10.1016/j.automatica.2006.05.006.

[14]

X. Liu and C. Xie, Control law in analytic expression of a system coupled by reaction-diffusion equation, Systems & Control Letters, 137 (2020), 104643, 5 pp. doi: 10.1016/j.sysconle.2020.104643.

[15]

F. MazencL. Praly and W. P. Dayawansa, Global stabilization by output feedback : Examples and counter-examples, Systems & Control Letters, 23 (1994), 119-125.  doi: 10.1016/0167-6911(94)90041-8.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

B. SaldivarS. Mondié and J. C. Avila Vilchis, The control of drilling vibrations: A coupled PDE-ODE modeling approach, International Journal Applied Mathematics Computer Science, 26 (2016), 335-349.  doi: 10.1515/amcs-2016-0024.

[18]

A. Smyshlyaev and M. Krstic, Closed-form boundary state feedbacks for a class of $1-d$ partial integro-differential equations, IEEE Transaction on Automatic Control, 49 (2004), 2185-2202.  doi: 10.1109/TAC.2004.838495.

[19]

G. A. Susto and M. Krstic, Control of PDE-ODE cascades with neumann interconnections, Journal of Franklin Institute, 347 (2010), 284-314.  doi: 10.1016/j.jfranklin.2009.09.005.

[20]

S. Tang and C. Xie, Stabilization for a coupled PDE-ODE control system, Journal of the Franklin Institute, 348 (2011), 2142-2155.  doi: 10.1016/j.jfranklin.2011.06.008.

[21]

S. Tang and C. Xie, State and output feedback boundary control for a coupled PDE-ODE system, Systems & Control Letters, 60 (2011), 540-545.  doi: 10.1016/j.sysconle.2011.04.011.

[22]

Y. Tang and G. Mazanti, Stability analysis of coupled linear ODE-hyperbolic PDE systems with two time scales, Automatica, 85 (2017), 386-396.  doi: 10.1016/j.automatica.2017.07.052.

[23]

G. Weiss and X. Zhao, Well-posedness and controllability of a class of coupled linear systems, SIAM Journal on Control and Optimization, 48 (2009), 2719-2750.  doi: 10.1137/090752833.

[24]

H.-N. Wu and J.-W. Wang, Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion pde-governed sensor dynamics, Nonlinear Dynamics, 72 (2013), 615-628.  doi: 10.1007/s11071-012-0740-4.

[25]

A. Zhao and C. Xie, Stabilization of coupled linear plant and reaction-diffusion process, Journal of the Franklin Institute, 351 (2014), 857-877.  doi: 10.1016/j.jfranklin.2013.09.012.

[26]

X. Zhao and G. Weiss, Controllability and observability of a well-posed system coupled with a finite-dimensional system, IEEE Transactions on Automatic Control, 56 (2011), 88-99.  doi: 10.1109/TAC.2010.2051352.

[27]

Z. Zhoua and S. Tang, Boundary stabilization of a coupled wave-ode system with internal anti-damping, International Journal of Control, 85 (2012), 1683-1693.  doi: 10.1080/00207179.2012.696704.

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