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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 & 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, (2003).   Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste România, (1976).   Google Scholar

[3]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, (1999).   Google Scholar

[4]

M. Boudaoud, Y. Haddab and Y. Le Gorrec, Modeling and optimal force control of a nonlinear electrostatic microgripper,, Mechatronics, 18 (2013), 1130.  doi: 10.1109/TMECH.2012.2197216.  Google Scholar

[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, (1998).   Google Scholar

[6]

B. Chentouf and J.-F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping,, ESAIM: Control, 4 (1999), 515.  doi: 10.1051/cocv:1999120.  Google Scholar

[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.  doi: 10.1081/PDE-120016132.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation,, J. Differential Equations, 198 (2004), 196.  doi: 10.1016/j.jde.2003.08.008.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.09.019.  Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer Monographs in Mathematics, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[12]

I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions,, J. Differential Equations, 254 (2013), 1741.  doi: 10.1016/j.jde.2012.11.009.  Google Scholar

[13]

F. Conrad and M. Pierre, Stabilization of Euler-Bernoulli Beam by Nonlinear Boundary Feedback,, Research Report RR-1235, (1990).   Google Scholar

[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.   Google Scholar

[15]

J.-M. Coron and B. d'Andrea Novel, Stabilization of a rotating body beam without damping,, Automatic Control, 43 (1998), 608.  doi: 10.1109/9.668828.  Google Scholar

[16]

J.-F. Couchouron, Compactness theorems for abstract evolution problems,, Journal of Evolution Equations, 2 (2002), 151.  doi: 10.1007/s00028-002-8084-z.  Google Scholar

[17]

M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets,, J. Functional Analysis, 3 (1969), 376.  doi: 10.1016/0022-1236(69)90032-9.  Google Scholar

[18]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups,, J. Functional Analysis, 13 (1973), 97.  doi: 10.1016/0022-1236(73)90069-4.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).   Google Scholar

[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), 2988.   Google Scholar

[21]

P. Grabowski, The motion planning problem and exponential stabilisation of a heavy chain. part i,, International Journal of Control, 82 (2009), 1539.  doi: 10.1080/00207170802588188.  Google Scholar

[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.  doi: 10.1080/01630560600657265.  Google Scholar

[23]

B. Guo, On the boundary control of a hybrid system with variable coefficients,, Journal of Optimization Theory and Applications, 114 (2002), 373.  doi: 10.1023/A:1016039819069.  Google Scholar

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs,, American Mathematical Society, (1988).   Google Scholar

[25]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar

[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, (2005), 351.  doi: 10.1007/11529798_22.  Google Scholar

[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.  doi: 10.1007/BF01766154.  Google Scholar

[28]

Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications,, Communications and Control Engineering Series, (1999).  doi: 10.1007/978-1-4471-0419-3.  Google Scholar

[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.  doi: 10.1007/s10440-014-9965-1.  Google Scholar

[30]

M. Miletić, D. Stürzer, A. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system,, Preprint, (2015).   Google Scholar

[31]

Ö. Morgül, Stabilization and disturbance rejection for the beam equation,, IEEE Transactions on Automatic Control, 46 (2001), 1913.  doi: 10.1109/9.975475.  Google Scholar

[32]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.   Google Scholar

[33]

A. Pazy, A class of semi-linear equations of evolution,, Israel J. Math., 20 (1975), 23.  doi: 10.1007/BF02756753.  Google Scholar

[34]

A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces,, J. Analyse Math., 40 (1981), 239.  doi: 10.1007/BF02790164.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

W. Rudin, Functional Analysis,, 2nd edition, (1991).   Google Scholar

[37]

D. Stürzer, Asymptitic Stability of an Euler-Bernoulli Beam Coupled to Nonlinear Controllers,, PhD thesis, (2015).   Google Scholar

[38]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences,, 2nd edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  doi: 10.1023/B:JODY.0000010063.69213.7c.  Google Scholar

[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.  doi: 10.1109/TAC.2008.2007176.  Google Scholar

[41]

G. F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces,, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 19.  doi: 10.1017/S0308210500016930.  Google Scholar

[42]

K. Yosida, Functional Analysis,, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar

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, (2003).   Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste România, (1976).   Google Scholar

[3]

G. K. Batchelor, An Introduction to Fluid Dynamics,, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, (1999).   Google Scholar

[4]

M. Boudaoud, Y. Haddab and Y. Le Gorrec, Modeling and optimal force control of a nonlinear electrostatic microgripper,, Mechatronics, 18 (2013), 1130.  doi: 10.1109/TMECH.2012.2197216.  Google Scholar

[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, (1998).   Google Scholar

[6]

B. Chentouf and J.-F. Couchouron, Nonlinear feedback stabilization of a rotating body-beam without damping,, ESAIM: Control, 4 (1999), 515.  doi: 10.1051/cocv:1999120.  Google Scholar

[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.  doi: 10.1081/PDE-120016132.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Global attractors for von Karman evolutions with a nonlinear boundary dissipation,, J. Differential Equations, 198 (2004), 196.  doi: 10.1016/j.jde.2003.08.008.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.09.019.  Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer Monographs in Mathematics, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[12]

I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups in supersonic flow-plate interactions,, J. Differential Equations, 254 (2013), 1741.  doi: 10.1016/j.jde.2012.11.009.  Google Scholar

[13]

F. Conrad and M. Pierre, Stabilization of Euler-Bernoulli Beam by Nonlinear Boundary Feedback,, Research Report RR-1235, (1990).   Google Scholar

[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.   Google Scholar

[15]

J.-M. Coron and B. d'Andrea Novel, Stabilization of a rotating body beam without damping,, Automatic Control, 43 (1998), 608.  doi: 10.1109/9.668828.  Google Scholar

[16]

J.-F. Couchouron, Compactness theorems for abstract evolution problems,, Journal of Evolution Equations, 2 (2002), 151.  doi: 10.1007/s00028-002-8084-z.  Google Scholar

[17]

M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets,, J. Functional Analysis, 3 (1969), 376.  doi: 10.1016/0022-1236(69)90032-9.  Google Scholar

[18]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups,, J. Functional Analysis, 13 (1973), 97.  doi: 10.1016/0022-1236(73)90069-4.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).   Google Scholar

[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), 2988.   Google Scholar

[21]

P. Grabowski, The motion planning problem and exponential stabilisation of a heavy chain. part i,, International Journal of Control, 82 (2009), 1539.  doi: 10.1080/00207170802588188.  Google Scholar

[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.  doi: 10.1080/01630560600657265.  Google Scholar

[23]

B. Guo, On the boundary control of a hybrid system with variable coefficients,, Journal of Optimization Theory and Applications, 114 (2002), 373.  doi: 10.1023/A:1016039819069.  Google Scholar

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs,, American Mathematical Society, (1988).   Google Scholar

[25]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar

[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, (2005), 351.  doi: 10.1007/11529798_22.  Google Scholar

[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.  doi: 10.1007/BF01766154.  Google Scholar

[28]

Z.-H. Luo, B.-Z. Guo and Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications,, Communications and Control Engineering Series, (1999).  doi: 10.1007/978-1-4471-0419-3.  Google Scholar

[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.  doi: 10.1007/s10440-014-9965-1.  Google Scholar

[30]

M. Miletić, D. Stürzer, A. Arnold and A. Kugi, Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system,, Preprint, (2015).   Google Scholar

[31]

Ö. Morgül, Stabilization and disturbance rejection for the beam equation,, IEEE Transactions on Automatic Control, 46 (2001), 1913.  doi: 10.1109/9.975475.  Google Scholar

[32]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.   Google Scholar

[33]

A. Pazy, A class of semi-linear equations of evolution,, Israel J. Math., 20 (1975), 23.  doi: 10.1007/BF02756753.  Google Scholar

[34]

A. Pazy, The Lyapunov method for semigroups of nonlinear contractions in Banach spaces,, J. Analyse Math., 40 (1981), 239.  doi: 10.1007/BF02790164.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

W. Rudin, Functional Analysis,, 2nd edition, (1991).   Google Scholar

[37]

D. Stürzer, Asymptitic Stability of an Euler-Bernoulli Beam Coupled to Nonlinear Controllers,, PhD thesis, (2015).   Google Scholar

[38]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences,, 2nd edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  doi: 10.1023/B:JODY.0000010063.69213.7c.  Google Scholar

[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.  doi: 10.1109/TAC.2008.2007176.  Google Scholar

[41]

G. F. Webb, Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces,, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 19.  doi: 10.1017/S0308210500016930.  Google Scholar

[42]

K. Yosida, Functional Analysis,, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar

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