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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 |
References:
[1] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Soc. Math., 306 (1988), 837.
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).
|
[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.
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), 108 (1987), 67.
|
[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.
doi: 10.1512/iumj.1995.44.2001. |
[6] |
T. Eisner, Stability of Operators and Operator Semigroups,, Operator Theory: Advances and Applications, (2010).
|
[7] |
K.-J. Engel, Generator property and stability for generalized difference operators,, J. Evol. Equ., 13 (2013), 311.
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, (2000).
doi: 10.1007/b97696. |
[9] |
L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces,, Trans. Amer. Math. Soc., 236 (1978), 385.
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.
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.
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, (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.
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.
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.
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.
|
[17] |
J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.
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., (2014).
doi: 10.1109/TAC.2014.2315754. |
[19] |
H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (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.
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, (1996).
doi: 10.1007/978-3-0348-9206-3. |
[22] |
J. A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems,, PhD thesis, (2007). Google Scholar |
[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.
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.
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.
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).
|
[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.
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), 108 (1987), 67.
|
[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.
doi: 10.1512/iumj.1995.44.2001. |
[6] |
T. Eisner, Stability of Operators and Operator Semigroups,, Operator Theory: Advances and Applications, (2010).
|
[7] |
K.-J. Engel, Generator property and stability for generalized difference operators,, J. Evol. Equ., 13 (2013), 311.
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, (2000).
doi: 10.1007/b97696. |
[9] |
L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces,, Trans. Amer. Math. Soc., 236 (1978), 385.
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.
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.
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, (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.
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.
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.
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.
|
[17] |
J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.
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., (2014).
doi: 10.1109/TAC.2014.2315754. |
[19] |
H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (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.
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, (1996).
doi: 10.1007/978-3-0348-9206-3. |
[22] |
J. A. Villegas, A port-Hamiltonian Approach to Distributed Parameter Systems,, PhD thesis, (2007). Google Scholar |
[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.
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.
doi: 10.1051/cocv/2009036. |
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