American Institute of Mathematical Sciences

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June  2014, 3(2): 207-229. doi: 10.3934/eect.2014.3.207

Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems

Björn Augner 1, and  Birgit Jacob 1,

1. 

Fachbereich C - Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany, Germany

Received  December 2013 Revised  April 2014 Published  May 2014

Stability and stabilization of linear port-Hamiltonian systems on infinite-dimensional spaces are investigated. This class is general enough to include models of beams and waves as well as transport and Schrödinger equations with boundary control and observation. The analysis is based on the frequency domain method which gives new results for second order port-Hamiltonian systems and hybrid systems. Stabilizing SIP or SOP controllers are designed. The obtained results are applied to the Euler-Bernoulli beam.
Keywords: $C_0$-semigroup, exponential stability, frequency domain method., Infinite-dimensional linear port-Hamiltonian systems, asymptotic stability, hybrid systems, stabilization.
Mathematics Subject Classification: Primary: 93D15, 93D20; Secondary: 35L25, 47D0.
Citation: Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207
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show all references

References:
[1]

Trans. Amer. Soc. Math., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[2]

Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986.  Google Scholar

[3]

SIAM J. Control Optim., 25 (1987), 526-546. doi: 10.1137/0325029.  Google Scholar

[4]

in Operator Methods for Optimal Control Problems (ed. S. J. Lee), Lecture Notes in Pure and Appl. Math., 108, Dekker, New York, 1987, 67-96.  Google Scholar

[5]

Indiana Univ. Math. J., 44 (1995), 545-573. doi: 10.1512/iumj.1995.44.2001.  Google Scholar

[6]

Operator Theory: Advances and Applications, 209, Birkhäuser Verlag, Basel, 2010.  Google Scholar

[7]

J. Evol. Equ., 13 (2013), 311-334. doi: 10.1007/s00028-013-0179-1.  Google Scholar

[8]

Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[9]

Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.  Google Scholar

[10]

SIAM J. Control Optim., 43 (2004), 341-356. doi: 10.1137/S0363012901380961.  Google Scholar

[11]

Systems Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006.  Google Scholar

[12]

Operator Theory: Advances and Applications, 223, Linear Operators and Linear Systems, Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0399-1.  Google Scholar

[13]

SIAM J. Control Optim., 44 (2005), 1864-1892. doi: 10.1137/040611677.  Google Scholar

[14]

Ann. Math. Pura Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.  Google Scholar

[15]

J. Comp. Appl. Math., 114 (2000), 1-10. doi: 10.1016/S0377-0427(99)00284-8.  Google Scholar

[16]

Studia Math., 88 (1988), 37-42.  Google Scholar

[17]

Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.  Google Scholar

[18]

IFAC Workshop on Control of Sys. Modeled by Part. Diff. Equ., CPDE, 2014. Available from: http://hal.archives-ouvertes.fr/hal-00872199. doi: 10.1109/TAC.2014.2315754.  Google Scholar

[19]

Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[20]

J. Geom. Phys., 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3.  Google Scholar

[21]

Operator Theory: Advances and Applications, 88, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3.  Google Scholar

[22]

PhD thesis, Universiteit Twente in Enschede, 2007. Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf. Google Scholar

[23]

IEEE Trans. Automat. Control, 54 (2009), 142-147. doi: 10.1109/TAC.2008.2007176.  Google Scholar

[24]

ESAIM Contr. Optim. Calc. Var., 16 (2010), 1077-1093. doi: 10.1051/cocv/2009036.  Google Scholar

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