Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback
Authors
Abstract
Boundary feedback stabilisation of linear port-Hamiltonian systems on an interval is considered. Generation and stability results already known for linear feedback are extended to nonlinear dissipative feedback, both to static feedback control and dynamic control via an (exponentially stabilising) nonlinear controller. A design method for nonlinear controllers of linear port-Hamiltonian systems is introduced. As a special case the Euler-Bernoulli beam is considered.
Citation
- Journal: SIAM Journal on Control and Optimization
- Year: 2019
- Volume: 57
- Issue: 3
- Pages: 1818–1844
- Publisher: Society for Industrial & Applied Mathematics (SIAM)
- DOI: 10.1137/15m1024901
BibTeX
@article{Augner_2019,
title={{Well-Posedness and Stability of Infinite-Dimensional Linear Port-Hamiltonian Systems with Nonlinear Boundary Feedback}},
volume={57},
ISSN={1095-7138},
DOI={10.1137/15m1024901},
number={3},
journal={SIAM Journal on Control and Optimization},
publisher={Society for Industrial & Applied Mathematics (SIAM)},
author={Augner, Björn},
year={2019},
pages={1818--1844}
}
References
- 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
- Calvert, B. & Gustafson, K. Multiplicative perturbation of nonlinear m-accretive operators. Journal of Functional Analysis vol. 10 149–158 (1972) – 10.1016/0022-1236(72)90046-8
- Conrad F., Berlin (1990)
- Cox, S. & Zuazua, E. The rate at which energy decays in a string damped at one end. Indiana University Mathematics Journal vol. 44 0–0 (1995) – 10.1512/iumj.1995.44.2001
- Chen, G., Delfour, M. C., Krall, A. M. & Payre, G. Modeling, Stabilization and Control of Serially Connected Beams. SIAM Journal on Control and Optimization vol. 25 526–546 (1987) – 10.1137/0325029
- Chen G., New York (1987)
- Dafermos, C. M. & Slemrod, M. Asymptotic behavior of nonlinear contraction semigroups. Journal of Functional Analysis vol. 13 97–106 (1973) – 10.1016/0022-1236(73)90069-4
- Engel, K.-J. Generator property and stability for generalized difference operators. Journal of Evolution Equations vol. 13 311–334 (2013) – 10.1007/s00028-013-0179-1
- Feng, D., Shi, D. & Zhang, W. Boundary feedback stabilization of Timoshenko beam with boundary dissipation. Science in China Series A: Mathematics vol. 41 483–490 (1998) – 10.1007/bf02879936
- 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
- Ramirez, H., Zwart, H. & Le Gorrec, Y. Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control. Automatica vol. 85 61–69 (2017) – 10.1016/j.automatica.2017.07.045
- Rauch, J. & Taylor, M. Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains. Indiana University Mathematics Journal vol. 24 79–86 (1974) – 10.1512/iumj.1975.24.24004
- Trostorff, S. A characterization of boundary conditions yielding maximal monotone operators. Journal of Functional Analysis vol. 267 2787–2822 (2014) – 10.1016/j.jfa.2014.08.009
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- van der Schaft A. J., Archiv für Elektronik und Übertragungstechnik (1995)
- 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
- 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