Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems

Authors

Lu Zhang, Fu Zheng, Sizhe Wang, Zhongjie Han

Abstract

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

Citation

Journal: Mathematical Methods in the Applied Sciences
Year: 2025
Volume: 48
Issue: 3
Pages: 3272–3290
Publisher: Wiley
DOI: 10.1002/mma.10484

BibTeX

@article{Zhang_2024,
  title={{Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems}},
  volume={48},
  ISSN={1099-1476},
  DOI={10.1002/mma.10484},
  number={3},
  journal={Mathematical Methods in the Applied Sciences},
  publisher={Wiley},
  author={Zhang, Lu and Zheng, Fu and Wang, Sizhe and Han, Zhongjie},
  year={2024},
  pages={3272--3290}
}

Download the bib file

References

Baozhu Guo & Cheng-Zhong Xu. On the spectrum-determined growth condition of a vibration cable with a tip mass. IEEE Transactions on Automatic Control vol. 45 89–93 (2000) – 10.1109/9.827360
Mei, Z.-D. Output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. Journal of Mathematical Analysis and Applications vol. 508 125860 (2022) – 10.1016/j.jmaa.2021.125860
Morgul, O., Bo Peng Rao & Conrad, F. On the stabilization of a cable with a tip mass. IEEE Transactions on Automatic Control vol. 39 2140–2145 (1994) – 10.1109/9.328811
Terrand-Jeanne, A., Andrieu, V., Tayakout-Fayolle, M. & Dos Santos Martins, V. Regulation of Inhomogeneous Drilling Model With a P-I Controller. IEEE Transactions on Automatic Control vol. 65 58–71 (2020) – 10.1109/tac.2019.2907792
Vanspranghe, N., Ferrante, F. & Prieur, C. Velocity Stabilization of a Wave Equation With a Nonlinear Dynamic Boundary Condition. IEEE Transactions on Automatic Control vol. 67 6786–6793 (2022) – 10.1109/tac.2021.3136086
DOI not foun – 10.1007/s10444‐004‐7629‐9
Kress R.. Numerical analysis graduate texts in mathematics (1998)
Guo, B.-Z. & Xu, B.-B. A semi-discrete finite difference method to uniform stabilization of wave equation with local viscosity. IFAC Journal of Systems and Control vol. 13 100100 (2020) – 10.1016/j.ifacsc.2020.100100
Liu, J. & Guo, B.-Z. A New Semidiscretized Order Reduction Finite Difference Scheme for Uniform Approximation of One-Dimensional Wave Equation. SIAM Journal on Control and Optimization vol. 58 2256–2287 (2020) – 10.1137/19m1246535
Liu, J. & Guo, B.-Z. A novel semi-discrete scheme preserving uniformly exponential stability for an Euler–Bernoulli beam. Systems & Control Letters vol. 134 104518 (2019) – 10.1016/j.sysconle.2019.104518
Liu, J., Hao, R. & Guo, B.-Z. Order reduction-based uniform approximation of exponential stability for one-dimensional Schrödinger equation. Systems & Control Letters vol. 160 105136 (2022) – 10.1016/j.sysconle.2022.105136
Freitas, P. & Zuazua, E. Stability Results for the Wave Equation with Indefinite Damping. Journal of Differential Equations vol. 132 338–352 (1996) – 10.1006/jdeq.1996.0183
Guo, B.-Z. & Jin, F.-F. Arbitrary decay rate for two connected strings with joint anti-damping by boundary output feedback. Automatica vol. 46 1203–1209 (2010) – 10.1016/j.automatica.2010.03.019
Hassine, F. Rapid Exponential Stabilization of a 1‐D Transmission Wave Equation with In‐domain Anti‐damping. Asian Journal of Control vol. 19 2017–2027 (2017) – 10.1002/asjc.1509
Macchelli, A., Gorrec, Y. L., Wu, Y. & Ramírez, H. Energy-based Control of a Wave Equation with Boundary Anti-damping. IFAC-PapersOnLine vol. 53 7740–7745 (2020) – 10.1016/j.ifacol.2020.12.1527
Smyshlyaev, A. & Krstic, M. Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Systems & Control Letters vol. 58 617–623 (2009) – 10.1016/j.sysconle.2009.04.005
Zhang, Y.-L., Zhu, M., Li, D. & Wang, J.-M. Stabilization of two coupled wave equations with joint anti-damping and non-collocated control. Automatica vol. 135 109995 (2022) – 10.1016/j.automatica.2021.109995
Infante, J. A. & Zuazua, E. Boundary observability for the space semi-discretizations of the 1 – d wave equation. ESAIM: Mathematical Modelling and Numerical Analysis vol. 33 407–438 (1999) – 10.1051/m2an:1999123
Zuazua, E. Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods. SIAM Review vol. 47 197–243 (2005) – 10.1137/s0036144503432862
Banks, H. T., Ito, K. & Wang, C. Exponentially stable approximations of weakly damped wave equations. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique 1–33 (1991) doi:10.1007/978-3-0348-6418-3_1 – 10.1007/978-3-0348-6418-3_1
DOI not foun – 10.1007/s00211‐002‐0442‐9
Liu, Z. & Zheng, S. Uniform Exponential Stability and Approximation in Control of a Thermoelastic System. SIAM Journal on Control and Optimization vol. 32 1226–1246 (1994) – 10.1137/s0363012991219006
Zheng, F. & Zhou, H. State reconstruction of the wave equation with general viscosity and non-collocated observation and control. Journal of Mathematical Analysis and Applications vol. 502 125257 (2021) – 10.1016/j.jmaa.2021.125257
Guo, B.-Z. & Zheng, F. Uniform Exponential Stability for a Schrödinger Equation and Its Semidiscrete Approximation. IEEE Transactions on Automatic Control vol. 69 8900–8907 (2024) – 10.1109/tac.2024.3419847
Wang, X., Xue, W., He, Y. & Zheng, F. Uniformly exponentially stable approximations for Timoshenko beams. Applied Mathematics and Computation vol. 451 128028 (2023) – 10.1016/j.amc.2023.128028
Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
Liu Z. Y.. Semigroups associated with dissipative systems (1999)
Abdallah, F., Nicaise, S., Valein, J. & Wehbe, A. Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications. ESAIM: Control, Optimisation and Calculus of Variations vol. 19 844–887 (2013) – 10.1051/cocv/2012036
DOI not foun – 10.1090/s0025‐5718‐98‐00915‐6