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November  2015, 20(9): 3029-3055. doi: 10.3934/dcdsb.2015.20.3029

## An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip

 1 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8, 1040 Vienna, Austria, Austria 2 Institute for Analysis and Scienti c Computing, Vienna University of Technology, Wiedner Hauptstraße 8, 1040 Vienna, Austria

Received  November 2014 Revised  July 2015 Published  September 2015

We study the asymptotic behavior for a system consisting of a clamped flexible beam that carries a tip payload, which is attached to a nonlinear damper and a nonlinear spring at its end. Characterizing the $\omega$-limit sets of the trajectories, we give a sufficient condition under which the system is asymptotically stable. In the case when this condition is not satisfied, we show that the beam deflection approaches a non-decaying time-periodic solution.
Citation: Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976, Translated from the Romanian. [3] G. K. Batchelor, An Introduction to Fluid Dynamics, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999. [4] M. Boudaoud, Y. Haddab and Y. Le Gorrec, Modeling and optimal force control of a nonlinear electrostatic microgripper, Mechatronics, IEEE/ASME Transactions on, 18 (2013), 1130-1139. doi: 10.1109/TMECH.2012.2197216. [5] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998. [6] B. Chentouf and J.-F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 515-535. doi: 10.1051/cocv:1999120. [7] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. [8] I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differential Equations, 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. [9] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86. doi: 10.1016/j.jde.2006.09.019. [10] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183pp. doi: 10.1090/memo/0912. [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics. doi: 10.1007/978-0-387-87712-9. [12] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Differential Equations, 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. [13] F. Conrad and M. Pierre, Stabilization of Euler-Bernoulli Beam by Nonlinear Boundary Feedback, Research Report RR-1235, 1990. http://hal.inria.fr/inria-00075324. [14] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 485-515. [15] J.-M. Coron and B. d'Andrea Novel, Stabilization of a rotating body beam without damping, Automatic Control, IEEE Transactions on, 43 (1998), 608-618. doi: 10.1109/9.668828. [16] J.-F. Couchouron, Compactness theorems for abstract evolution problems, Journal of Evolution Equations, 2 (2002), 151-175. doi: 10.1007/s00028-002-8084-z. [17] M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418. doi: 10.1016/0022-1236(69)90032-9. [18] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis, 13 (1973), 97-106. doi: 10.1016/0022-1236(73)90069-4. [19] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [20] S. S. Ge, S. Zhang and W. He, Modeling and control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, in American Control Conference (ACC), 2011, IEEE, 2011, 2988-2993. [21] P. Grabowski, The motion planning problem and exponential stabilisation of a heavy chain. part i, International Journal of Control, 82 (2009), 1539-1563. doi: 10.1080/00207170802588188. [22] B.-Z. Guo and J.-M. Wang, Riesz basis generation of abstract second-order partial differential equation systems with general non-separated boundary conditions, Numerical Functional Analysis and Optimization, 27 (2006), 291-328. doi: 10.1080/01630560600657265. [23] B. Guo, On the boundary control of a hybrid system with variable coefficients, Journal of Optimization Theory and Applications, 114 (2002), 373-395. doi: 10.1023/A:1016039819069. [24] J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. [25] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. [26] A. Kugi and D. Thull, Infinite-dimensional decoupling control of the tip position and the tip angle of a composite piezoelectric beam with tip mass, in Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, vol. 322 of Lecture Notes in Control and Information Science, Springer Berlin Heidelberg, 2005, 351-368. doi: 10.1007/11529798_22. [27] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4), 152 (1988), 281-330. doi: 10.1007/BF01766154. [28] Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series, Springer-Verlag London Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3. [29] M. Miletić and A. Arnold, A piezoelectric Euler-Bernoulli beam with dynamic boundary control: Stability and dissipative FEM, Acta Applicandae Mathematicae, 138 (2015), 241-277. doi: 10.1007/s10440-014-9965-1. [30] M. Miletić, D. Stürzer, A. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system, Preprint, 2015, arXiv:1505.07576. [31] Ö. Morgül, Stabilization and disturbance rejection for the beam equation, IEEE Transactions on Automatic Control, 46 (2001), 1913-1918. doi: 10.1109/9.975475. [32] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [33] A. Pazy, A class of semi-linear equations of evolution, Israel J. Math., 20 (1975), 23-36. doi: 10.1007/BF02756753. [34] A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. Analyse Math., 40 (1981), 239-262 (1982). doi: 10.1007/BF02790164. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. [37] D. Stürzer, Asymptitic Stability of an Euler-Bernoulli Beam Coupled to Nonlinear Controllers, PhD thesis, Vienna University of Technology, To appear in 2015. [38] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [39] H. R. Thieme and I. I. Vrabie, Relatively compact orbits and compact attractors for a class of nonlinear evolution equations, J. Dynam. Differential Equations, 15 (2003), 731-750. doi: 10.1023/B:JODY.0000010063.69213.7c. [40] J. A. Villegas, H. Zwart, Y. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Transactions on Automatic Control, 54 (2009), 142-147. doi: 10.1109/TAC.2008.2007176. [41] G. F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 19-33. doi: 10.1017/S0308210500016930. [42] K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976, Translated from the Romanian. [3] G. K. Batchelor, An Introduction to Fluid Dynamics, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999. [4] M. Boudaoud, Y. Haddab and Y. Le Gorrec, Modeling and optimal force control of a nonlinear electrostatic microgripper, Mechatronics, IEEE/ASME Transactions on, 18 (2013), 1130-1139. doi: 10.1109/TMECH.2012.2197216. [5] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998. [6] B. Chentouf and J.-F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 515-535. doi: 10.1051/cocv:1999120. [7] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. [8] I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation, J. Differential Equations, 198 (2004), 196-231. doi: 10.1016/j.jde.2003.08.008. [9] I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86. doi: 10.1016/j.jde.2006.09.019. [10] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183pp. doi: 10.1090/memo/0912. [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010, Well-posedness and long-time dynamics. doi: 10.1007/978-0-387-87712-9. [12] I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions, J. Differential Equations, 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009. [13] F. Conrad and M. Pierre, Stabilization of Euler-Bernoulli Beam by Nonlinear Boundary Feedback, Research Report RR-1235, 1990. http://hal.inria.fr/inria-00075324. [14] F. Conrad and M. Pierre, Stabilization of second order evolution equations by unbounded nonlinear feedback, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 485-515. [15] J.-M. Coron and B. d'Andrea Novel, Stabilization of a rotating body beam without damping, Automatic Control, IEEE Transactions on, 43 (1998), 608-618. doi: 10.1109/9.668828. [16] J.-F. Couchouron, Compactness theorems for abstract evolution problems, Journal of Evolution Equations, 2 (2002), 151-175. doi: 10.1007/s00028-002-8084-z. [17] M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418. doi: 10.1016/0022-1236(69)90032-9. [18] C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis, 13 (1973), 97-106. doi: 10.1016/0022-1236(73)90069-4. [19] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. [20] S. S. Ge, S. Zhang and W. He, Modeling and control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, in American Control Conference (ACC), 2011, IEEE, 2011, 2988-2993. [21] P. Grabowski, The motion planning problem and exponential stabilisation of a heavy chain. part i, International Journal of Control, 82 (2009), 1539-1563. doi: 10.1080/00207170802588188. [22] B.-Z. Guo and J.-M. Wang, Riesz basis generation of abstract second-order partial differential equation systems with general non-separated boundary conditions, Numerical Functional Analysis and Optimization, 27 (2006), 291-328. doi: 10.1080/01630560600657265. [23] B. Guo, On the boundary control of a hybrid system with variable coefficients, Journal of Optimization Theory and Applications, 114 (2002), 373-395. doi: 10.1023/A:1016039819069. [24] J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. [25] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. [26] A. Kugi and D. Thull, Infinite-dimensional decoupling control of the tip position and the tip angle of a composite piezoelectric beam with tip mass, in Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, vol. 322 of Lecture Notes in Control and Information Science, Springer Berlin Heidelberg, 2005, 351-368. doi: 10.1007/11529798_22. [27] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4), 152 (1988), 281-330. doi: 10.1007/BF01766154. [28] Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series, Springer-Verlag London Ltd., London, 1999. doi: 10.1007/978-1-4471-0419-3. [29] M. Miletić and A. Arnold, A piezoelectric Euler-Bernoulli beam with dynamic boundary control: Stability and dissipative FEM, Acta Applicandae Mathematicae, 138 (2015), 241-277. doi: 10.1007/s10440-014-9965-1. [30] M. Miletić, D. Stürzer, A. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system, Preprint, 2015, arXiv:1505.07576. [31] Ö. Morgül, Stabilization and disturbance rejection for the beam equation, IEEE Transactions on Automatic Control, 46 (2001), 1913-1918. doi: 10.1109/9.975475. [32] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [33] A. Pazy, A class of semi-linear equations of evolution, Israel J. Math., 20 (1975), 23-36. doi: 10.1007/BF02756753. [34] A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. Analyse Math., 40 (1981), 239-262 (1982). doi: 10.1007/BF02790164. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. [37] D. Stürzer, Asymptitic Stability of an Euler-Bernoulli Beam Coupled to Nonlinear Controllers, PhD thesis, Vienna University of Technology, To appear in 2015. [38] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [39] H. R. Thieme and I. I. Vrabie, Relatively compact orbits and compact attractors for a class of nonlinear evolution equations, J. Dynam. Differential Equations, 15 (2003), 731-750. doi: 10.1023/B:JODY.0000010063.69213.7c. [40] J. A. Villegas, H. Zwart, Y. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Transactions on Automatic Control, 54 (2009), 142-147. doi: 10.1109/TAC.2008.2007176. [41] G. F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 19-33. doi: 10.1017/S0308210500016930. [42] K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
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