Exponential Decay Rate of Linear Port-Hamiltonian Systems: A Multiplier Approach

Authors

Luis A. Mora, Kirsten Morris

Abstract

In this work, the multiplier method is extended to obtain a general lower bound on the exponential decay rate for port-Hamiltonian systems in one space dimension with boundary dissipation. The physical parameters of the system may be spatially varying. It is shown that, under assumptions of boundary or internal dissipation, the system is exponentially stable. This is established through a Lyapunov function defined through a general multiplier function. Furthermore, an explicit bound on the decay rate in terms of the physical parameters is obtained. The method is applied to a number of examples.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2024
• Volume: 69
• Issue: 3
• Pages: 1767–1772
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2023.3332008

BibTeX

@article{Mora_2024,
  title={{Exponential Decay Rate of Linear Port-Hamiltonian Systems: A Multiplier Approach}},
  volume={69},
  ISSN={2334-3303},
  DOI={10.1109/tac.2023.3332008},
  number={3},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Mora, Luis A. and Morris, Kirsten},
  year={2024},
  pages={1767--1772}
}

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References

• Komornik, Exact Controllability and Stabilization, The Multiplier Method (1994)
• Russell, D. L. & Weiss, G. A General Necessary Condition for Exact Observability. SIAM Journal on Control and Optimization vol. 32 1–23 (1994) – 10.1137/s036301299119795x
• Xu, C.-Z. Exact observability and exponential stability of infinite-dimensional bilinear systems. Mathematics of Control, Signals, and Systems vol. 9 73–93 (1996) – 10.1007/bf01211519
• Zuazua, E. A remark on the observability of conservative linear systems. Contemporary Mathematics 47–59 (2012) doi:10.1090/conm/577/11462 – 10.1090/conm/577/11462
• 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
• Yan, L. & Sun, L. General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term. Mathematical Methods in the Applied Sciences vol. 46 879–894 (2022) – 10.1002/mma.8554
• Cheng, Y., Wu, Y. & Guo, B.-Z. Boundary Stability Criterion for a Nonlinear Axially Moving Beam. IEEE Transactions on Automatic Control vol. 67 5714–5729 (2022) – 10.1109/tac.2021.3124754
• Rivera, J. E. M., Racke, R., Sepúlveda, M. & Villagrán, O. V. On Exponential Stability for Thermoelastic Plates: Comparison and Singular Limits. Applied Mathematics & Optimization vol. 84 1045–1081 (2020) – 10.1007/s00245-020-09670-7
• Guesmia, A. A New Approach of Stabilization of Nondissipative Distributed Systems. SIAM Journal on Control and Optimization vol. 42 24–52 (2003) – 10.1137/s0363012901394978
• Guo, F. & Huang, F. Boundary Feedback Stabilization of the Undamped Euler–Bernoulli Beam with Both Ends Free. SIAM Journal on Control and Optimization vol. 43 341–356 (2004) – 10.1137/s0363012901380961
• Wu, Y., Xue, X. & Shen, T. Absolute stability of the Kirchhoff string with sector boundary control. Automatica vol. 50 1915–1921 (2014) – 10.1016/j.automatica.2014.05.006
• Tucsnak, M. & Weiss, G. Observation and Control for Operator Semigroups. (Birkhäuser Basel, 2009). doi:10.1007/978-3-7643-8994-9 – 10.1007/978-3-7643-8994-9
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Villegas, Boundary control for a class of dissipative differential operators including diffusion systems. Proc. 17th Int. Symp. Math. Theory Netw. Syst. (2006)
• Zwart, H., Le Gorrec, Y., Maschke, B. & Villegas, J. Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: Control, Optimisation and Calculus of Variations vol. 16 1077–1093 (2009) – 10.1051/cocv/2009036
• Villegas, J. A., Zwart, H., Le Gorrec, Y. & Maschke, B. Exponential Stability of a Class of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 54 142–147 (2009) – 10.1109/tac.2008.2007176
• Augner, B. & Jacob, B. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory vol. 3 207–229 (2014) – 10.3934/eect.2014.3.207
• Macchelli, A. On the control by interconnection and exponential stabilisation of infinite dimensional port-Hamiltonian systems. 2016 IEEE 55th Conference on Decision and Control (CDC) 3137–3142 (2016) doi:10.1109/cdc.2016.7798739 – 10.1109/cdc.2016.7798739
• Trostorff, Characterisation of exponential stability for port-Hamiltonian systems. (2022)
• Mattioni, A., Wu, Y. & Le Gorrec, Y. A Lyapunov Approach for the Exponential Stability of a Damped Timoshenko Beam. IEEE Transactions on Automatic Control vol. 68 8287–8292 (2023) – 10.1109/tac.2023.3297499
• Mora, L. A. & Morris, K. Exponential Decay Rate of port-Hamiltonian Systems with one side Boundary Damping. IFAC-PapersOnLine vol. 55 400–405 (2022) – 10.1016/j.ifacol.2022.11.086
• Mora, L. A. & Morris, K. Exponential decay rate bound of one-dimensional distributed port-Hamiltonian systems with boundary dissipation. 2022 IEEE 61st Conference on Decision and Control (CDC) 409–414 (2022) doi:10.1109/cdc51059.2022.9993139 – 10.1109/cdc51059.2022.9993139
• Serhani, A., Matignon, D. & Haine, G. Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations. IFAC-PapersOnLine vol. 52 96–101 (2019) – 10.1016/j.ifacol.2019.08.017
• Zimmer, B. J., Lipshitz, S. P., Morris, K. A., Vanderkooy, J. & Obasi, E. E. An Improved Acoustic Model for Active Noise Control in a Duct. Journal of Dynamic Systems, Measurement, and Control vol. 125 382–395 (2003) – 10.1115/1.1592192
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Mattioni, A., Wu, Y. & Le Gorrec, Y. Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach. Applied Mathematical Modelling vol. 83 59–75 (2020) – 10.1016/j.apm.2020.02.008
• Morris, K. A. & Özer, A. Ö. Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects. SIAM Journal on Control and Optimization vol. 52 2371–2398 (2014) – 10.1137/130918319