EXPONENTIAL STABILITY OF SAINT-VENANT EQUATIONS IN SUPERCRITICAL FLOW

Jingwen Wang; Dongxia Zhao

doi:10.11948/20250198

2026 Volume 16 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(3): 1244-1255

Next Article Previous Article

Jingwen Wang, Dongxia Zhao. EXPONENTIAL STABILITY OF SAINT-VENANT EQUATIONS IN SUPERCRITICAL FLOW[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1244-1255. doi: 10.11948/20250198

Citation:

Jingwen Wang, Dongxia Zhao. EXPONENTIAL STABILITY OF SAINT-VENANT EQUATIONS IN SUPERCRITICAL FLOW[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1244-1255. doi: 10.11948/20250198

EXPONENTIAL STABILITY OF SAINT-VENANT EQUATIONS IN SUPERCRITICAL FLOW

Jingwen Wang^1,,
Dongxia Zhao^1, ,

1.
School of Mathematics, North University of China, Taiyuan 030051, China

Author Bio: Email: 1500068705@qq.com(J. Wang)
Corresponding author: Email: zhaodongxia6@sina.com(D. Zhao)

Abstract

Abstract

In this paper, the stability of a class of $2\times 2$ hyperbolic systems with the same sign propagation speeds is investigated under proportional feedback control. Semigroup theory is used to verify the well-posedness of the system. The exponential stability of the closed-loop system is analyzed by constructing a strictly weighted Lyapunov function. In addition, the characteristic equation and eigenfunctions of the system are deduced. For the special case where $\gamma$ is equal to 0, by using Schur-Cohn criterion, the sufficient and necessary stability conditions, which make the eigenvalues have the negative real part, is obtained.
- Saint-Venant equations /
- supercritical flow /
- Lyapunov function /
- exponential stability
MSC: 34H05, 35L60, 93D15

FullText(HTML)

References

[1]	G. A. de Andrade, R. Vazquez, I. Karafyllis and M. Krstic, Backstepping control of a hyperbolic PDE system with zero characteristic speed states, IEEE Trans. Autom. Control, 2024, 69(10), 6988–6995. Google Scholar
[2]	G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Birkhäuser Cham, Switzerland, 2016. Google Scholar
[3]	G. Bastin and J.-M. Coron, Exponential stability of PI control for Saint-Venant equations with a friction term, Methods Appl. Anal., 2019, 26(2), 101–112. Google Scholar
[4]	G. Bastin, J.-M. Coron and B. d'Andréa Novel, On lyapunov stability of linearised Saint-Venant equations for a sloping, Netw. Heterog. Media, 2009, 4(2), 177–187. Google Scholar
[5]	G. Bastin, J.-M. Coron and A. Hayat, Feedforward boundary control of 2×2 nonlinear hyperbolic systems with application to Saint-Venant equations, Eur. J. Control., 2021, 57, 41–53. Google Scholar
[6]	R. Benzaid and A. Benaissa. Indirect boundary stabilization for weakly coupled degenerate wave equations under fractional damping, J. Appl. Anal. Comput., 2024, 14(3), 1735–1770. Google Scholar
[7]	J. Deutscher, N. Gehring and N. Jung, Backstepping control of coupled general hyperbolic-parabolic PDE-PED systems, IEEE Trans. Autom. Control, 2023, 69(5), 3465–3472. Google Scholar
[8]	A. Diagne, M. Diagne, S. Tang and M. Krstic, Backstepping stabilization of the linearized Saint-Venant–Exner model, Automatica, 2017, 76, 345–354. Google Scholar
[9]	M. Gugat, G. Leugering and E. J. P. G. Schmidt, Global controllability between steady supercritical flows in channel networks, Math. Methods Appl. Sci., 2004, 27(7), 781–802. Google Scholar
[10]	B. Guo, Y. Xiao and Y. Ban, Orbital stability of solitary waves for the nonlinear Schrödinger-KdV system, J. Appl. Anal. Comput., 2022, 12(1), 245–255. Google Scholar
[11]	A. Hayat, On boundary stability of inhomogeneous 2×2 1-D hyperbolic systems for the C¹ norm, ESAIM: Control Optim. Calc. Var., 2019, 25(82). Google Scholar
[12]	A. Hayat, Y. Hu and P. Shang, PI control for the cascade channels modeled by general Saint-Venant equations, IEEE Trans. Autom. Control, 2023, 69(8), 4974–4987. Google Scholar
[13]	A. Hayat and P. Shang, A quadratic lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope, Automatica, 2019, 100, 52–60. Google Scholar
[14]	J. P. LaSalle, The Stability and Control of Discrete Processes, Springer-Verlag, New York, 1986. Google Scholar
[15]	E. Somathilake and M. Diagne, Output feedback control of suspended sediment load entrainment in water canals and reservoirs, IFAC-PapersOnLine, 2024, 58(28), 983–988. Google Scholar
[16]	H. Yu and M. Krstic, Traffic Congestion Control by PDE Backstepping, Birkhäuser Cham, Switzerland, 2022. Google Scholar
[17]	B. Zhang, F. Zheng and Y. He, Uniformly exponentially stable approximation for the transmission line with variable coefficients and its application, J. Appl. Anal. Comput., 2024, 14(4), 2228–2256. Google Scholar
[18]	D. Zhao and F. Bao, The stability analysis for a 2×2 conservation law with PI controller and PDP controller, J. Math., 2023, 2023, 4382517. Google Scholar
[19]	D. Zhao, D. Fan and Y. Guo, The spectral analysis and exponential stability of a 1-d 2×2 hyperbolic system with proportional feedback control, Int. J. Control, Autom. Syst., 2022, 20(8), 2633–2640. Google Scholar