Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian Systems

Authors

Abstract

We study the non-autonomous version of an infinite-dimensional linear port-Hamiltonian system on an interval [ a , b ] . Employing results on evolution families, we show C 1 -well-posedness of the corresponding Cauchy problem, and thereby existence and uniqueness of classical solutions for sufficiently regular initial data. Further, we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non-autonomous setting.

Keywords

evolution family, infinite-dimensional port-hamiltonian system, non-autonomous cauchy problem, uniform exponential stability, well-posedness

Citation

Journal: Systems & Control Letters
Year: 2020
Volume: 144
Issue:
Pages: 104757
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2020.104757

BibTeX

@article{Augner_2020,
  title={{Exponential stability for infinite-dimensional non-autonomous port-Hamiltonian Systems}},
  volume={144},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2020.104757},
  journal={Systems \&amp; Control Letters},
  publisher={Elsevier BV},
  author={Augner, Björn and Laasri, Hafida},
  year={2020},
  pages={104757}
}

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References

Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30(4). https://doi.org/10.1007/s00498-018-0223- – 10.1007/s00498-018-0223-3
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/10.1137/04061167 – 10.1137/040611677
Jacob, Linear port-hamiltonian systems on infinite-dimensional spaces. (2012)
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.2014.3.20 – 10.3934/eect.2014.3.207
Villegas, (2007)
Zwart H, Le Gorrec Y, Maschke B, Villegas J (2009) Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16(4):1077–1093. https://doi.org/10.1051/cocv/200903 – 10.1051/cocv/2009036
Villegas JA, Le Gorrec Y, Zwart H, van der Schaft AJ (2005) BOUNDARY CONTROL SYSTEMS AND THE SYSTEM NODE. IFAC Proceedings Volumes 38(1):308–313. https://doi.org/10.3182/20050703-6-cz-1902.0062 – 10.3182/20050703-6-cz-1902.00622
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- – 10.1007/s00028-014-0271-1
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.2008.200717 – 10.1109/tac.2008.2007176
Engel, (2000)
Schnaubelt R, Weiss G (2010) Two classes of passive time-varying well-posed linear systems. Math Control Signals Syst 21(4):265–301. https://doi.org/10.1007/s00498-010-0049- – 10.1007/s00498-010-0049-0
Augner A, Jacob B, Laasri H (2015) On the right multiplicative perturbation of non-autonomous $L^p$-maximal regularity. J Operator Theory 74(2):391–415. https://doi.org/10.7900/jot.2014jul31.206 – 10.7900/jot.2014jul31.2064
Volterra, (1938)
KATO T (1953) Integration of the equation of evolution in a Banach space. J Math Soc Japan 5(2). https://doi.org/10.2969/jmsj/0052020 – 10.2969/jmsj/00520208
Kato, Linear evolution equations of hyperbolic type. J. Fac. Sci. Univ. Tokyo (1970)
Tanabe, (1979)
Sobolevskii, Equations of parabolic type in a Banach space. Trudy Moskov. Mat. Obsc. (1961)
Acquistapace, A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Univ. Padova (1987)
Howland JS (1974) Stationary scattering theory for time-dependent Hamiltonians. Math Ann 207(4):315–335. https://doi.org/10.1007/bf0135134 – 10.1007/bf01351346
Chicone, Evolution semigroups in dynamical systems and differential equations. (1999)
Nagel, Wellposedness for nonautonomous abstract Cauchy problems. (2002)
Schnaubelt, Well-posedness and asymptotic behaviour of nonautonomous evolution equation. (2002)
Lions, (1961)
Lions, (1972)
Batty CJK, Chill R, Tomilov Y (2002) Strong Stability of Bounded Evolution Families and Semigroups. Journal of Functional Analysis 193(1):116–139. https://doi.org/10.1006/jfan.2001.391 – 10.1006/jfan.2001.3917
Haraux A (1983) Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes. Journal of Differential Equations 49(3):473–483. https://doi.org/10.1016/0022-0396(83)90008- – 10.1016/0022-0396(83)90008-6
Paunonen L, Seifert D (2019) Asymptotics for periodic systems. Journal of Differential Equations 266(11):7152–7172. https://doi.org/10.1016/j.jde.2018.11.02 – 10.1016/j.jde.2018.11.028
Cox S, Zuazua E (1995) The rate at which energy decays in a string damped at one end. Indiana Univ Math J 44(2):0–0. https://doi.org/10.1512/iumj.1995.44.200 – 10.1512/iumj.1995.44.2001
Nickel, (1996)