Boundary feedback stabilization of quasilinear hyperbolic systems with zero characteristic speed
Abstract
In this paper, we investigate the boundary feedback stabilization of a quasilinear hyperbolic system with zero characteristic speed and a partially dissipative structure. This structure enables us to construct a Lyapunov function that guarantees exponential stability for the H2 solution. We also introduce another set of stability conditions by restricting terms corresponding to zero eigenvalues to the dissipative part, which still ensures exponential stability. As an application, we achieve feedback stabilization for the modified model of neurofilament transport in axons.
Keyword : quasilinear hyperbolic system, zero characteristic speed, feedback stabilization, Lyapunov function
How to Cite
Wang, Z., & Yao, W. (2025). Boundary feedback stabilization of quasilinear hyperbolic systems with zero characteristic speed. Mathematical Modelling and Analysis, 30(2), 299–321. https://doi.org/10.3846/mma.2025.20890
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T. Li. Global Classical Solutions for Quasilinear Hyperbolic Systems, volume 32 of RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
K. Wang, Z. Wang and W. Yao. Boundary feedback stabilization of quasilinear hyperbolic systems with partially dissipative structure. Systems & Control Lett., 146:104815, 9, 2020. https://doi.org/10.1016/j.sysconle.2020.104815
Z. Wang. Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Ann. Math. Ser. B, 27(6):643–656, 2006. https://doi.org/10.1007/s11401-005-0520-2
C.-Z. Xu and G. Sallet. Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM Control Optim. Calc. Var., 7:421–442, 2002. https://doi.org/10.1051/cocv:2002062
W.-A. Yong. Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differential Equations, 155(1):89–132, 1999. https://doi.org/10.1006/jdeq.1998.3584
W.-A. Yong. Boundary stabilization of hyperbolic balance laws with characteristic boundaries. Automatica J. IFAC, 101:252–257, 2019. https://doi.org/10.1016/j.automatica.2018.12.003
G. Bastin and J.-M. Coron. Stability and Boundary Stabilization of 1-D Hyperbolic Systems, volume 88 of Progress in Nonlinear Differential Equations and their Applications, Subseries in Control. Birkha¨user/Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-32062-5
Y. Chitour, G. Mazanti and M. Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Netw. Heterog. Media, 11(4):563–601, 2016. https://doi.org/10.3934/nhm.2016010
J.-M. Coron. Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007.
J.-M. Coron, G. Bastin and B. d’Andr´ea Novel. Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems. SIAM J. Control Optim., 47(3):1460–1498, 2008. https://doi.org/10.1137/070706847
J.-M. Coron, O. Glass and Z. Wang. Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed. SIAM J. Control Optim., 48(5):3105–3122, 2009. https://doi.org/10.1137/090749268
J.-M. Coron and H.-M. Nguyen. Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems. SIAM J. Math. Anal., 47(3):2220–2240, 2015. https://doi.org/10.1137/140976625
J.-M. Coron and H.-M. Nguyen. Optimal time for the controllability of linear hyperbolic systems in one-dimensional space. SIAM J. Control Optim., 57(2):1127– 1156, 2019. https://doi.org/10.1137/18M1185600
J.-M. Coron and H.-M. Nguyen. Null-controllability of linear hyperbolic systems in one dimensional space. Systems & Control Lett., 148:Paper No. 104851, 8, 2021. https://doi.org/10.1016/j.sysconle.2020.104851
J.-M. Coron, R. Vazquez, M. Krstic and G. Bastin. Local exponential H2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J. Control Optim., 51(3):2005–2035, 2013. https://doi.org/10.1137/120875739
G. Craciun, A. Brown and A. Friedman. A dynamical system model of neurofilament transport in axons. J. Theoret. Biol., 237(3):316–322, 2005. https://doi.org/10.1016/j.jtbi.2005.04.018
M. Dick, M. Gugat and G. Leugering. A strict H1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numer. Algebra Control Optim., 1(2):225–244, 2011. https://doi.org/10.3934/naco.2011.1.225
J.M. Greenberg and T. Li. The effect of boundary damping for the quasilinear wave equation. J. Differential Equations, 52(1):66–75, 1984. https://doi.org/10.1016/0022-0396(84)90135-9
M. Gugat, G. Leugering and K. Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: a strict H2-Lyapunov function. Math. Control Relat. Fields, 7(3):419–448, 2017. https://doi.org/10.3934/mcrf.2017015
A. Gustavo, R. Vazquez, I. Karafyllis and M. Krstic. Backstepping control of a hyperbolic PDE system with zero characteristic speed. IFAC-PapersOnLine, 55(26):168–173, 2022. ISSN 2405-8963. https://doi.org/10.1016/j.ifacol.2022.10.395
A. Hayat. Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the C1 norm. SIAM J. Control Optim., 57(6):3603–3638, 2019. https://doi.org/10.1137/17M1150803
M. Herty and W.-A. Yong. Feedback boundary control of linear hyperbolic systems with relaxation. Automatica J. IFAC, 69:12–17, 2016. ISSN 0005-1098. https://doi.org/10.1016/j.automatica.2016.02.016
L. Hu, R. Vazquez, F. Di Meglio and M. Krstic. Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems. SIAM J. Control Optim., 57(2):963–998, 2019. https://doi.org/10.1137/15M1012712
L. Hu and Z. Wang. On boundary control of a hyperbolic system with a vanishing characteristic speed. ESAIM Control Optim. Calc. Var., 22(1):134–147, 2016. https://doi.org/10.1051/cocv/2015031
X. Huang, Z. Wang and S. Zhou. The dichotomy property in stabilizability of 2 × 2 linear hyperbolic systems. ESAIM Control Optim. Calc. Var., 30:Paper No. 24, 24, 2024. https://doi.org/10.1051/cocv/2024010
M. Krstic and A. Smyshlyaev. Boundary control of PDEs: A course on backstepping designs, volume 16 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. https://doi.org/10.1137/1.9780898718607
T. Li. Global Classical Solutions for Quasilinear Hyperbolic Systems, volume 32 of RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
K. Wang, Z. Wang and W. Yao. Boundary feedback stabilization of quasilinear hyperbolic systems with partially dissipative structure. Systems & Control Lett., 146:104815, 9, 2020. https://doi.org/10.1016/j.sysconle.2020.104815
Z. Wang. Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Ann. Math. Ser. B, 27(6):643–656, 2006. https://doi.org/10.1007/s11401-005-0520-2
C.-Z. Xu and G. Sallet. Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM Control Optim. Calc. Var., 7:421–442, 2002. https://doi.org/10.1051/cocv:2002062
W.-A. Yong. Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differential Equations, 155(1):89–132, 1999. https://doi.org/10.1006/jdeq.1998.3584
W.-A. Yong. Boundary stabilization of hyperbolic balance laws with characteristic boundaries. Automatica J. IFAC, 101:252–257, 2019. https://doi.org/10.1016/j.automatica.2018.12.003