Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control


We herein investigate the boundary input-to-state stability (ISS) of a class of coupled hyperbolic partial differential equation-ordinary differential equation (PDE-ODE) systems with respect to the presence of uncertainties and external disturbances. The boundary feedback control of the proportional type acts on the ODE part and indirectly affects the hyperbolic PDE dynamics via the boundary input. Using the strict Lyapunov function, some sufficient conditions in terms of matrix inequalities are obtained for the boundary ISS of the closed-loop hyperbolic PDE-ODE systems. The feedback control laws are designed by combining the line search algorithm and polytopic embedding techniques. The effectiveness of the designed boundary control is assessed by applying it to the system of interconnected continuous stirred tank reactor and a plug flow reactor through a numerical simulation.

This is a preview of subscription content, access via your institution.


  1. 1

    Hasan A, Aamo O M, Krstic M. Boundary observer design for hyperbolic PDE-ODE cascade systems. Automatica, 2016, 68: 75–86

    MathSciNet  Article  Google Scholar 

  2. 2

    Tang Y, Prieur C, Girard A. Stability analysis of a singularly perturbed coupled ODE-PDE system. In: Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, 2015. 4591–4596

    Google Scholar 

  3. 3

    Zhou H C, Guo B Z. Performance output tracking for one-dimensional wave equation subject to unmatched general disturbance and non-collocated control. Eur J Contr, 2017, 39: 39–52

    MathSciNet  Article  Google Scholar 

  4. 4

    Zhang L G, Prieur C. Necessary and sufficient conditions on the exponential stability of positive hyperbolic systems. IEEE Trans Automat Contr, 2017, 62: 3610–3617

    MathSciNet  Article  Google Scholar 

  5. 5

    Zhang L G, Prieur C, Qiao J F. Local exponential stabilization of semi-linear hyperbolic systems by means of a boundary feedback control. IEEE Control Syst Lett, 2018, 2: 55–60

    Article  Google Scholar 

  6. 6

    Zhang L G, Prieur C. Stochastic stability of Markov jump hyperbolic systems with application to traffic flow control. Automatica, 2017, 86: 29–37

    MathSciNet  Article  Google Scholar 

  7. 7

    Diagne M, Bekiaris-Liberis N, Krstic M. Time- and state-dependent input delay-compensated bang-bang control of a screw extruder for 3D printing. Int J Robust Nonlin, 2017, 27: 3727–3757

    MathSciNet  MATH  Google Scholar 

  8. 8

    Moghadam A A, Aksikas I, Dubljevic S, et al. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica, 2013, 49: 526–533

    MathSciNet  Article  Google Scholar 

  9. 9

    Krstic M, Smyshlyaev A. Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst Control Lett, 2008, 57: 750–758

    MathSciNet  Article  Google Scholar 

  10. 10

    Krstic M. Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst Control Lett, 2009, 58: 372–377

    MathSciNet  Article  Google Scholar 

  11. 11

    Li J, Liu Y G. Stabilization of coupled pde-ode systems with spatially varying coefficient. J Syst Sci Complex, 2013, 26: 151–174

    MathSciNet  Article  Google Scholar 

  12. 12

    Karafyllis I, Jiang Z P. Stability and Stabilization of Nonlinear Systems. London: Springer-Verlag, 2011

    Google Scholar 

  13. 13

    Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201

    MathSciNet  Article  Google Scholar 

  14. 14

    Yang C D, Cao J D, Huang T W, et al. Guaranteed cost boundary control for cluster synchronization of complex spatio-temporal dynamical networks with community structure. Sci China Inf Sci, 2018, 61: 052203

    MathSciNet  Article  Google Scholar 

  15. 15

    Ito H, Dashkovskiy S, Wirth F. Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems. Automatica, 2012, 48: 1197–1204

    MathSciNet  Article  Google Scholar 

  16. 16

    Geiselhart R, Wirth F. Numerical construction of LISS Lyapunov functions under a small gain condition. In: Proceedings of the 50th IEEE Conference on Decision and Control, Orlando, 2012. 25–30

    Google Scholar 

  17. 17

    Dashkovskiy S, Rüffer B S, Wirth F R. An ISS small gain theorem for general networks. Math Control Signals Syst, 2007, 19: 93–122

    MathSciNet  Article  Google Scholar 

  18. 18

    Dashkovskiy S, Mironchenko A. Input-to-state stability of infinite-dimensional control systems. Math Control Signals Syst, 2013, 25: 1–35

    MathSciNet  Article  Google Scholar 

  19. 19

    Karafyllis I, Krstic M. On the relation of delay equations to first-order hyperbolic partial differential equations. Esaim Control Optim Calc Var, 2013, 20: 894–923

    MathSciNet  Article  Google Scholar 

  20. 20

    Prieur C, Mazenc F. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math Control Signals Syst, 2012, 24: 111–134

    MathSciNet  Article  Google Scholar 

  21. 21

    Tanwani A, Prieur C, Tarbouriech S. Input-to-state stabilization in H1-norm for boundary controlled linear hyperbolic PDEs with application to quantized control. In: Proceedings of the 55th IEEE Conference on Decision and Control, Vegas, 2016. 3112–3117

    Google Scholar 

  22. 22

    Espitia N, Girard A, Marchand N, et al. Fluid-flow modeling and stability analysis of communication networks. In: Proceedings of the 20th IFAC World Congress, Toulouse, 2017. 4534–4539

    Google Scholar 

  23. 23

    Karafyllis I, Krstic M. ISS in different norms for 1-D parabolic PDES with boundary disturbances. SIAM J Control Optim, 2017, 55: 1716–1751

    MathSciNet  Article  Google Scholar 

  24. 24

    Bastin G, Coron J M. Stability and boundary stabilization of 1-D hyperbolic systems. In: Progress in Nonlinear Differential Equations and Their Applications. Berlin: Springer, 2016

    Google Scholar 

  25. 25

    Aksikas I, Winkin J J, Dochain D. Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans Automat Contr, 2007, 52: 1179–1193

    MathSciNet  Article  Google Scholar 

  26. 26

    Shampine L F. Solving hyperbolic PDEs in MATLAB. Appl Num Anal Comp Math, 2005, 2: 346–358

    MathSciNet  Article  Google Scholar 

Download references


This work was supported by National Natural Science Foundation of China (Grant Nos. 61374076, 61533002) and Beijing Municipal Natural Science Foundation (Grant No. 1182001).

Author information



Corresponding author

Correspondence to Liguo Zhang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Hao, J. & Qiao, J. Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control. Sci. China Inf. Sci. 62, 42201 (2019).

Download citation


  • hyperbolic PDE-ODE systems
  • input-to-state stability
  • boundary control
  • Lyapunov function
  • PFR-CSTR models