# American Institute of Mathematical Sciences

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 & 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. doi: 10.1090/S0002-9947-1988-0933321-3. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] T. Eisner, Stability of Operators and Operator Semigroups,, Operator Theory: Advances and Applications, (2010). Google Scholar [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. Google Scholar [8] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000). doi: 10.1007/b97696. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 37. Google Scholar [17] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar [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. Google Scholar [19] H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] T. Eisner, Stability of Operators and Operator Semigroups,, Operator Theory: Advances and Applications, (2010). Google Scholar [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. Google Scholar [8] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000). doi: 10.1007/b97696. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 37. Google Scholar [17] J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847. doi: 10.2307/1999112. Google Scholar [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. Google Scholar [19] H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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