Stability of the multidimensional wave equation in port-Hamiltonian modelling

Authors

Birgit Jacob, Nathanael Skrepek

Abstract

We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian fashion and show that the system is semi-uniformly stable, which is a stability concept between exponential stability and strong stability. Hence, this also implies strong stability of the system. In particular, classical solutions are uniformly stable. This will be achieved by showing that the spectrum of the port-Hamiltonian operator is contained in the left half plane C− and the port-Hamiltonian operator generates a contraction semigroup. Moreover, we show that the spectrum consists of eigenvalues only and the port-Hamiltonian operator has a compact resolvent.

Citation

• Journal: 2021 60th IEEE Conference on Decision and Control (CDC)
• Year: 2021
• Volume:
• Issue:
• Pages: 6188–6193
• Publisher: IEEE
• DOI: 10.1109/cdc45484.2021.9683501

BibTeX

@inproceedings{Jacob_2021,
  title={{Stability of the multidimensional wave equation in port-Hamiltonian modelling}},
  DOI={10.1109/cdc45484.2021.9683501},
  booktitle={{2021 60th IEEE Conference on Decision and Control (CDC)}},
  publisher={IEEE},
  author={Jacob, Birgit and Skrepek, Nathanael},
  year={2021},
  pages={6188--6193}
}

Download the bib file

References

• Jeltsema, D. & Van Der Schaft, A. J. Lagrangian and Hamiltonian formulation of transmission line systems with boundary energy flow. Reports on Mathematical Physics vol. 63 55–74 (2009) – 10.1016/s0034-4877(09)00009-3
• kurula, Linear wave systems on n-D spatial domains. Internat J Control (2015)
• Lagnese, J. Decay of solutions of wave equations in a bounded region with boundary dissipation. Journal of Differential Equations vol. 50 163–182 (1983) – 10.1016/0022-0396(83)90073-6
• 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
• Macchelli, A. & Melchiorri, C. Control by interconnection of mixed port Hamiltonian systems. IEEE Transactions on Automatic Control vol. 50 1839–1844 (2005) – 10.1109/tac.2005.858656
• Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes vol. 25 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
• Monk, P. Finite Element Methods for Maxwell’s Equations. (2003) doi:10.1093/acprof:oso/9780198508885.001.0001 – 10.1093/acprof:oso/9780198508885.001.0001
• Quinn, J. P. & Russell, D. L. Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proceedings of the Royal Society of Edinburgh: Section A Mathematics vol. 77 97–127 (1977) – 10.1017/s0308210500018072
• schaft, Port-Hamiltonian systems: an introductory survey. Proc the International Congress of Mathematicians (2006)
• 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
• dautray, Mathematical Analysis and Numerical Methods for Science and Technology (1990)
• dautray, Mathematical Analysis and Numerical Methods for Science and Technology (1990)
• engel, One-Parameter Semigroups for Linear Evolution Equations (2000)
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Humaloja, J.-P., Kurula, M. & Paunonen, L. Approximate Robust Output Regulation of Boundary Control Systems. IEEE Transactions on Automatic Control vol. 64 2210–2223 (2019) – 10.1109/tac.2018.2884676
• evans, Partial Differential Equations (2010)
• Chill, R., Seifert, D. & Tomilov, Y. Semi-uniform stability of operator semigroups and energy decay of damped waves. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 378 20190614 (2020) – 10.1098/rsta.2019.0614
• 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
• Bardos, C., Lebeau, G. & Rauch, J. Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary. SIAM Journal on Control and Optimization vol. 30 1024–1065 (1992) – 10.1137/0330055
• skrepek, Well-posedness of linear first order port-Hamiltonian systems on multidimensional spatial domains. Evolution Equations and Control Theory (2020)
• Tao, X. & Zhang, S. Boundary unique continuation theorems under zero Neumann boundary conditions. Bulletin of the Australian Mathematical Society vol. 72 67–85 (2005) – 10.1017/s0004972700034882
• Su, P., Tucsnak, M. & Weiss, G. Stabilizability properties of a linearized water waves system. Systems & Control Letters vol. 139 104672 (2020) – 10.1016/j.sysconle.2020.104672
• yosida, Functional Analysis (1980)
• villegas, A Port-Hamiltonian Approach to Distributed Parameter Systems. PhD thesis (2007)
• 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