Asymptotic stability for a class of boundary control systems with non-linear damping

Authors

Hans Zwart, Hector Ramirez, Yann Le Gorrec

Abstract

The asymptotic stability of boundary controlled port-Hamiltonian systems defined on a 1D spatial domain interconnected to a class of non-linear boundary damping is addressed. It is shown that if the port-Hamiltonian system is approximately observable, then any boundary damping which behaves linear for small velocities asymptotically stabilizes the system.

Keywords

Boundary control systems; infinite-dimensional port Hamiltonian systems; asymptotic stability; non-linear control

Citation

• Journal: IFAC-PapersOnLine
• Year: 2016
• Volume: 49
• Issue: 8
• Pages: 304–308
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2016.07.458
• Note: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016- Bertinoro, Italy, 13—15 June 2016

BibTeX

@article{Zwart_2016,
  title={{Asymptotic stability for a class of boundary control systems with non-linear damping}},
  volume={49},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2016.07.458},
  number={8},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Zwart, Hans and Ramirez, Hector and Le Gorrec, Yann},
  year={2016},
  pages={304--308}
}

Download the bib file

References

• Augner B, Jacob B (2014) Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. EECT 3(2):207–229. https://doi.org/10.3934/eect.2014.3.207 – 10.3934/eect.2014.3.207
• Curtain, (1995)
• Curtain R, Zwart H (2016) Stabilization of collocated systems by nonlinear boundary control. Systems & Control Letters 96:11–14. https://doi.org/10.1016/j.sysconle.2016.06.014 – 10.1016/j.sysconle.2016.06.014
• Curtain, R. and Zwart, H. (2016). TBA. Springer-Verslag. In progress.
• Jacob, (2012)
• Jacob B, Morris K, Zwart H (2015) C 0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain. J Evol Equ 15(2):493–502. https://doi.org/10.1007/s00028-014-0271-1 – 10.1007/s00028-014-0271-1
• Le Gorrec, Y., Zwart, H., and Maschke, B. (2004). A semigroup approach to port hamiltonian systems associated with linear skew symmetric operator. 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2004).
• Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/040611677 – 10.1137/040611677
• Oostveen, (2000)
• Pazy, (1983)
• Ramirez, Exponential stabilization of boundary controlled port-Hamiltonian systems with dynamic feedback. Automatic Control, IEEE Transactions on (2014)
• Staffans, (2005)
• van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1-2):166–194. https://doi.org/10.1016/s0393-0440(01)00083-3 – 10.1016/s0393-0440(01)00083-3
• Villegas JA, Zwart H, Le Gorrec Y, Maschke B (2009) Exponential Stability of a Class of Boundary Control Systems. IEEE Trans Automat Contr 54(1):142–147. https://doi.org/10.1109/tac.2008.2007176 – 10.1109/tac.2008.2007176
• Villegas, Stability and stabilization of a class of boundary control systems. Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ‘05. 44th IEEE Conference on (2005)
• Villegas, (2007)
• Zwart, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: Control, Optimisation and Calculus of Variations (2010)