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

Previous Article
On controllability of a linear elastic beam with memory under longitudinal load
EECT Home
This Issue
Next Article

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

Abstract
Related Papers

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

References:

[1]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Soc. Math., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.
[2]	W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986.
[3]	G. Chen, M. C. Delfour, A. M. Krall and G. Payres, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546. doi: 10.1137/0325029.
[4]	G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The euler-bernoulli beam equation with boundary energy dissipation, 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.
[5]	S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573. doi: 10.1512/iumj.1995.44.2001.
[6]	T. Eisner, Stability of Operators and Operator Semigroups, Operator Theory: Advances and Applications, 209, Birkhäuser Verlag, Basel, 2010.
[7]	K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334. doi: 10.1007/s00028-013-0179-1.
[8]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.
[9]	L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.
[10]	F. Guo and F. Huang, Boundary feedback stabilization of the undamped Euler-Bernoulli beam with both ends free, SIAM J. Control Optim., 43 (2004), 341-356. doi: 10.1137/S0363012901380961.
[11]	B.-Z. Guo, J.-M. Wang and S.-P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Systems Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006.
[12]	B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, 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.
[13]	Y. Le Gorrec, H. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892. doi: 10.1137/040611677.
[14]	W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Math. Pura Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.
[15]	K. Liu and Z. Liu, Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip, J. Comp. Appl. Math., 114 (2000), 1-10. doi: 10.1016/S0377-0427(99)00284-8.
[16]	Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.
[17]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[18]	H. Ramirez, H. Zwart and Y. Le Gorrec, Exponential Stability of Boundary Controlled Port Hamiltonian Systems with Dynamic Feedback, 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.
[19]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[20]	A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow, J. Geom. Phys., 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3.
[21]	J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory: Advances and Applications, 88, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3.
[22]	J. A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems, PhD thesis, Universiteit Twente in Enschede, 2007. Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf.
[23]	J. A. Villegas, H. Zwart, Y. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Trans. Automat. Control, 54 (2009), 142-147. doi: 10.1109/TAC.2008.2007176.
[24]	H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Contr. Optim. Calc. Var., 16 (2010), 1077-1093. doi: 10.1051/cocv/2009036.

show all references

References:

[1]	W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Soc. Math., 306 (1988), 837-852. doi: 10.1090/S0002-9947-1988-0933321-3.
[2]	W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986.
[3]	G. Chen, M. C. Delfour, A. M. Krall and G. Payres, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546. doi: 10.1137/0325029.
[4]	G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The euler-bernoulli beam equation with boundary energy dissipation, 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.
[5]	S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573. doi: 10.1512/iumj.1995.44.2001.
[6]	T. Eisner, Stability of Operators and Operator Semigroups, Operator Theory: Advances and Applications, 209, Birkhäuser Verlag, Basel, 2010.
[7]	K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334. doi: 10.1007/s00028-013-0179-1.
[8]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.
[9]	L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.
[10]	F. Guo and F. Huang, Boundary feedback stabilization of the undamped Euler-Bernoulli beam with both ends free, SIAM J. Control Optim., 43 (2004), 341-356. doi: 10.1137/S0363012901380961.
[11]	B.-Z. Guo, J.-M. Wang and S.-P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Systems Control Lett., 54 (2005), 557-574. doi: 10.1016/j.sysconle.2004.10.006.
[12]	B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, 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.
[13]	Y. Le Gorrec, H. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892. doi: 10.1137/040611677.
[14]	W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Math. Pura Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.
[15]	K. Liu and Z. Liu, Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip, J. Comp. Appl. Math., 114 (2000), 1-10. doi: 10.1016/S0377-0427(99)00284-8.
[16]	Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.
[17]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[18]	H. Ramirez, H. Zwart and Y. Le Gorrec, Exponential Stability of Boundary Controlled Port Hamiltonian Systems with Dynamic Feedback, 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.
[19]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[20]	A. J. van der Schaft and B. M. Maschke, Hamiltonian formulation of distributed parameter systems with boundary energy flow, J. Geom. Phys., 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3.
[21]	J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory: Advances and Applications, 88, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3.
[22]	J. A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems, PhD thesis, Universiteit Twente in Enschede, 2007. Available from: http://doc.utwente.nl/57842/1/thesis_Villegas.pdf.
[23]	J. A. Villegas, H. Zwart, Y. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Trans. Automat. Control, 54 (2009), 142-147. doi: 10.1109/TAC.2008.2007176.
[24]	H. Zwart, Y. Le Gorrec, B. Maschke and J. Villegas, Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain, ESAIM Contr. Optim. Calc. Var., 16 (2010), 1077-1093. doi: 10.1051/cocv/2009036.

[1]	Jochen Schmid. Stabilization of port-Hamiltonian systems with discontinuous energy densities. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2021063
[2]	Nathanael Skrepek. Well-posedness of linear first order port-Hamiltonian Systems on multidimensional spatial domains. Evolution Equations and Control Theory, 2021, 10 (4) : 965-1006. doi: 10.3934/eect.2020098
[3]	Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066
[4]	Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control and Related Fields, 2022, 12 (1) : 245-273. doi: 10.3934/mcrf.2021021
[5]	Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[6]	Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control and Related Fields, 2022, 12 (1) : 17-47. doi: 10.3934/mcrf.2021001
[7]	Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073
[8]	Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations and Control Theory, 2022, 11 (1) : 199-224. doi: 10.3934/eect.2020108
[9]	Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243
[10]	Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734
[11]	Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149
[12]	Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83
[13]	Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758
[14]	Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577
[15]	Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[16]	Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[17]	Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069
[18]	Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[19]	Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072
[20]	Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks and Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267

2020 Impact Factor: 1.081

Tools

Metrics

Other articles
by authors

on AIMS
Björn Augner Birgit Jacob
on Google Scholar
Björn Augner Birgit Jacob

[Back to Top]