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

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

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.
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.

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