The Direct Feedback Control and Exponential Stabilization of a Coupled Heat PDE-ODE System with Dirichlet Boundary Interconnection

  • Dong-Xia ZhaoEmail author
  • Jun-Min Wang
  • Ya-Ping Guo
Regular Papers Control Theory and Applications


This paper addresses the exponential stability for an interconnected system of an nth-order ODE system with the input governed by the Dirichlet boundary of a heat equation, and conversely, the output of the ODE is fluxed into the heat equation. The semigroup approach is adopted to show that the system operator is well-posed. We establish the exponential stability of the system by Riesz basis method. Furthermore, with MATLAB software, some numerical simulations are presented to show the effectiveness of the interconnection between the heat PDE and ODE systems.


Coupled system exponential stability heat equation spectral analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. N. Singh, “Robust Nonlinear Attitude Control of flexible Spacecraft,” IEEE Transactions on Aerospace & Electronic System, vol. AES–23, no. 3, pp. 380–387, 2007.Google Scholar
  2. [2]
    C. Chalons, M. L. D. Monache, and P. Goatin, “A conservative scheme for non–classical solutions to a strongly coupled PDE–ODE problem,” Analysis of PDEs, 2014.Google Scholar
  3. [3]
    W. Omar, Y. W. Fu, J. M. Lai, and Q. L. Zhang, “Simulation of RF circuit with a PDE model of the MOSFET,” Proc. of 4th International Conference on ASIC, pp. 662–665, 2001.Google Scholar
  4. [4]
    F. M. Atay, “Balancing the inverted pendulum using position feedback,” Applied Mathematics Letters, vol. 12, pp. 51–56, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. M. Wang, X.W. Lv, and D. X. Zhao, “Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks,” International Journal of Control, vol. 84, no. 5, pp. 904–915, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. X. Zhao and J. M. Wang, “Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks,” Journal of Dynamical and Control Systems, vol. 18, no. 2, pp. 904–915, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    H. Khalil, “A note on the robustness of high–gain–observerbased controllers to unmodeled actuator and sensor dynamics,” Automatica, vol. 41, no. 10, pp. 1821–1824, 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    B. Liu and H. Y. Hu, “Stabilization of linear undamped systems via position and delayed position feedbacks,” Journal of Sound and Vibration, vol. 312, pp. 509–525, 2008.Google Scholar
  9. [9]
    K. Q. Gu and S. Niculescu, “Survey on recents rusults in the stability and control of time–delay systems,” Journal of Dynamic Systems, Measurement, and Control, vol. 125, pp. 158–165, 2003.CrossRefGoogle Scholar
  10. [10]
    J. J. Gu and J. M. Wang, “Backstepping state feedback regulator design for an unstable reaction–diffusion PDE with long time delay,” Journal of Dynamical and Control Systems, vol. 24, no. 4, pp. 563–576, 2018.MathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Krstic, “Compensating actuator and sensor dynamics governed by diffusion PDEs,” Systems & Control Letters, vol. 58, pp. 372–377, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. A. Susto and M. Krstic, “Control of PDE–ODE cascades with Neumann interconnections,” Journal of the Franklin Institute, vol. 347, pp. 284–314, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M. Krstic, “Compensating a string PDE in the actuation or in sensing path of an unstable ODE,” IEEE Transactions on Automatic Control, vol. 54, pp. 1362.1368, 2009.Google Scholar
  14. [14]
    B.B. Ren, J.M. Wang and Krstic M, “Stabilization of an ODE–Schrödinger cascade,” Systems & Control Letters, vol. 62, pp. 503–510, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. J. Gu, C. Q. Wei and J. M. Wang, “Backstepping–based output regulation of ODEs cascaded by wave equation with in–domain anti–damping,” Transactions of the Institute of Measurement and Control, in press, 2018. DOI: 10.1177/0142331217752040Google Scholar
  16. [16]
    J. W. Wang, H. N. Wu, and H. X. Li, “Static output feedback control design for linear MIMO systems with actuator dynamics governed by diffusion PDEs,” International Journal of Control, vol. 87, no. 1, pp. 90–100, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. M. Wang, L. L. Su, and H. X. Li, “Stabilization of an unstable reaction–diffusion PDE cascaded with a heat eqnation,” Systems & Control Letters, vol. 76, pp. 8–18, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. M. Wang, B. B. Ren, and M. Krstic, “Stabilization and Gevrey regularity of a Schrödinger eqnarray in boundary feedback with a heat eqnation,” IEEE Transactions on Automatic Control, vol. 57, no. 1, pp. 179–185, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L. Lu and J. M. Wang, “Stabilization of Schrödinger Equation in dynamic boundary feedback with a memory typed heat equation,” International Journal of Control, in press, 2018. DOI: 10.1080/00207179.2017.1358826Google Scholar
  20. [20]
    D. X. Zhao and J. M. Wang, “Stabilization of the pendulum system by coupling with a heat eqnation,” Journal of Vibration and Control, vol. 20, no. 16, pp. 2443–2449, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Eqnations, Springer–Verlag, New York, 1983.CrossRefzbMATHGoogle Scholar
  22. [22]
    B. Z. Guo, “Riesz basis approach to the stabilization of a flexible beam with a tip mass,” SIAM Journal on Control and Optimization, vol. 39, pp. 1736.1747, 2001.Google Scholar
  23. [23]
    M. R. Opmeer, “Nuclearity of Hankel operators for ultradifferentiable control systems,” Systems & Control Letters, vol. 57, pp. 913–918, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    X. Y. Liu, D. W. C. Ho, J. D. Cao, and W. Y. Xu, “Discontinuous Observers Design for Finite–Time Consensus of Multiagent Systems With External Disturbances,” IEEE Transactions on Neural Networks&Learning Systems, vol. 28, no. 11, pp. 2826–2830, 2017.MathSciNetCrossRefGoogle Scholar
  25. [25]
    X. Y. Liu, J. D. Cao, W. W. Yu, and Q. Song, “Nonsmooth Finite–Time Synchronization of Switched Coupled Neural Networks,” IEEE Transactions on Cybernetics, vol. 46, no. 10, pp. 2360–2371, 2016.CrossRefGoogle Scholar
  26. [26]
    X. Y. Liu, D. W. C. Ho, Q. Song, and J. D. Cao, “Finite–/fixed–time robust stabilization of switched discontinuous systems with disturbances,” Nonlinear Dynamics, vol. 90, no. 3, pp. 2057–2068, 2017.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNorth University of ChinaTaiyuanChina
  2. 2.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  3. 3.School of MathematicsShanxi UniversityTaiyuanChina

Personalised recommendations