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Erschienen in: Journal of Applied Mathematics and Computing 1-2/2016

01.06.2016 | Original Research

Boundary control of nonlinear elastic systems

verfasst von: K. D. Do

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2016

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Abstract

This paper presents a design of boundary controllers for global stabilization of nonlinear elastic systems, which cover nonlinear elastic strings and membranes, under external bounded forces. The boundary controllers guarantee exponential convergence of the unique system solution to a ball centered at the origin. The Faedo–Galerkin approximation method is used to prove existence and uniqueness of the solution of the closed-loop system. The control design is based on the Lyapunov direct method, Gronwall’s, Poincare’s, and Holder’s inequalities, and Sobolev embedding theorems. Simulations illustrate the effectiveness of the proposed controllers.

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Literatur
1.
Zurück zum Zitat Ono, K.: Global existence, decay, and blow up of solutions for some mldly degenerate nonlinear Kirchhoff strings. J. Differ. Equ. 137, 273–301 (1997)CrossRefMATH Ono, K.: Global existence, decay, and blow up of solutions for some mldly degenerate nonlinear Kirchhoff strings. J. Differ. Equ. 137, 273–301 (1997)CrossRefMATH
2.
3.
Zurück zum Zitat Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 88, 1–10 (2004)MathSciNetMATH Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 88, 1–10 (2004)MathSciNetMATH
4.
Zurück zum Zitat Cavalcanti, M.M., Cavalcanti, V.N.D., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)MathSciNetMATH Cavalcanti, M.M., Cavalcanti, V.N.D., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1–14 (2002)MathSciNetMATH
5.
Zurück zum Zitat Alves, C.O., Cavalcanti, M.M.: On existence, uniform decay rates and blow up for solutions of the 2-d wave equation with exponential source. Calc. Var. Part. Differ. Equ. 34, 377–411 (2009)MathSciNetCrossRefMATH Alves, C.O., Cavalcanti, M.M.: On existence, uniform decay rates and blow up for solutions of the 2-d wave equation with exponential source. Calc. Var. Part. Differ. Equ. 34, 377–411 (2009)MathSciNetCrossRefMATH
6.
Zurück zum Zitat Aassila, M., Cavalcanti, M.M., Soriano, J.A.: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim. 38, 1581–1602 (2000)MathSciNetCrossRefMATH Aassila, M., Cavalcanti, M.M., Soriano, J.A.: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim. 38, 1581–1602 (2000)MathSciNetCrossRefMATH
7.
Zurück zum Zitat Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Dover Publications, Mineola (1990)MATH Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Dover Publications, Mineola (1990)MATH
8.
Zurück zum Zitat Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995)MATH Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995)MATH
9.
Zurück zum Zitat Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)MATH Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)MATH
10.
Zurück zum Zitat Balas, M.J.: Active control of flexible systems. In: Proceeding of the AIAA Symposium on Dynamic and Control of Large Flexible Spacecraft, vol. 23, pp. 217–236. (1977) Balas, M.J.: Active control of flexible systems. In: Proceeding of the AIAA Symposium on Dynamic and Control of Large Flexible Spacecraft, vol. 23, pp. 217–236. (1977)
12.
Zurück zum Zitat Meirovitch, L.: Principles and Techniques of Vibrations. Prentice-Hall, Upper Saddle River (1997) Meirovitch, L.: Principles and Techniques of Vibrations. Prentice-Hall, Upper Saddle River (1997)
13.
Zurück zum Zitat Ge, S.S., Lee, T.H., Zhu, G.: A nonlinear feedback controller for a single-link flexible manipulator based on a finite element model. J. Robot. Syst. 14(3), 165–178 (1997)CrossRefMATH Ge, S.S., Lee, T.H., Zhu, G.: A nonlinear feedback controller for a single-link flexible manipulator based on a finite element model. J. Robot. Syst. 14(3), 165–178 (1997)CrossRefMATH
14.
Zurück zum Zitat Gawronski, W.: Dynamics and Control of Structures a Modal Approach. Springer, New York (1998)CrossRefMATH Gawronski, W.: Dynamics and Control of Structures a Modal Approach. Springer, New York (1998)CrossRefMATH
15.
Zurück zum Zitat Ravindran, S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Methods Fluids 34(5), 425–448 (2000)MathSciNetCrossRefMATH Ravindran, S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Methods Fluids 34(5), 425–448 (2000)MathSciNetCrossRefMATH
16.
Zurück zum Zitat Lyapunov, A.M.: Stability of Motion. Academic Press, New York (1966)MATH Lyapunov, A.M.: Stability of Motion. Academic Press, New York (1966)MATH
17.
Zurück zum Zitat Shahruz, S.M.: Boundary control of the axially moving Kirchhoff string. Automatica 34, 1273–1277 (1998)CrossRefMATH Shahruz, S.M.: Boundary control of the axially moving Kirchhoff string. Automatica 34, 1273–1277 (1998)CrossRefMATH
18.
Zurück zum Zitat Yang, K.-J., Hong, K.-S., Matsuno, F.: Robust adaptive boundary control of an axially moving string under a spatiotemporally varying tension. J. Sound Vib. 273, 1007–1029 (2004)MathSciNetCrossRefMATH Yang, K.-J., Hong, K.-S., Matsuno, F.: Robust adaptive boundary control of an axially moving string under a spatiotemporally varying tension. J. Sound Vib. 273, 1007–1029 (2004)MathSciNetCrossRefMATH
19.
Zurück zum Zitat Fung, R.F., Wu, J.M., Wu, S.L.: Stabilization of an axially moving string by nonlinear boundary feedback. J. Dyn. Syst. Meas. Control 121, 117–121 (1999)CrossRefMATH Fung, R.F., Wu, J.M., Wu, S.L.: Stabilization of an axially moving string by nonlinear boundary feedback. J. Dyn. Syst. Meas. Control 121, 117–121 (1999)CrossRefMATH
20.
Zurück zum Zitat Fung, R.F., Tseng, C.C.: Boundary control of an axially moving string via Lyapunov method. J. Dyn. Syst. Meas. Control 121, 105–110 (1999)CrossRef Fung, R.F., Tseng, C.C.: Boundary control of an axially moving string via Lyapunov method. J. Dyn. Syst. Meas. Control 121, 105–110 (1999)CrossRef
21.
Zurück zum Zitat Queiroz, M.S.D., Dawson, M., Nagarkatti, S., Zhang, F.: Lyapunov-Based Control of Mechanical Systems. Birkhauser, Boston (2000)CrossRefMATH Queiroz, M.S.D., Dawson, M., Nagarkatti, S., Zhang, F.: Lyapunov-Based Control of Mechanical Systems. Birkhauser, Boston (2000)CrossRefMATH
22.
Zurück zum Zitat Liu, K., Liu, Z.: Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip. J. Comput. Appl. Math. 114, 1–10 (2000)MathSciNetMATH Liu, K., Liu, Z.: Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip. J. Comput. Appl. Math. 114, 1–10 (2000)MathSciNetMATH
23.
Zurück zum Zitat Fard, M.P., Sagatun, S.I.: Exponential stabilization of a transversely vibrating beam by boundary control via Lyapunov’s direct method. J. Dyn. Syst. Meas. Control 123, 195–200 (2001)CrossRefMATH Fard, M.P., Sagatun, S.I.: Exponential stabilization of a transversely vibrating beam by boundary control via Lyapunov’s direct method. J. Dyn. Syst. Meas. Control 123, 195–200 (2001)CrossRefMATH
24.
Zurück zum Zitat Tanaka, N., Iwamoto, H.: Active boundary control of an Euler–Bernoulli beam for generating vibration-free state. J. Sound Vib. 340, 570–586 (2007)MathSciNetCrossRefMATH Tanaka, N., Iwamoto, H.: Active boundary control of an Euler–Bernoulli beam for generating vibration-free state. J. Sound Vib. 340, 570–586 (2007)MathSciNetCrossRefMATH
25.
Zurück zum Zitat Do, K.D., Pan, J.: Boundary control of transverse motion of marine risers with actuator dynamics. J. Sound Vib. 318(4–5), 768–791 (2008)CrossRef Do, K.D., Pan, J.: Boundary control of transverse motion of marine risers with actuator dynamics. J. Sound Vib. 318(4–5), 768–791 (2008)CrossRef
26.
Zurück zum Zitat Do, K.D., Pan, J.: Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009)CrossRef Do, K.D., Pan, J.: Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009)CrossRef
27.
Zurück zum Zitat Ge, S.S., He, W., How, B.V., Choo, Y.S.: Boundary control of a coupled nonlinear flexible marine riser. IEEE Trans. Control Syst.Technol. 18(5), 1080–1091 (2010)CrossRef Ge, S.S., He, W., How, B.V., Choo, Y.S.: Boundary control of a coupled nonlinear flexible marine riser. IEEE Trans. Control Syst.Technol. 18(5), 1080–1091 (2010)CrossRef
28.
Zurück zum Zitat Do, K.D.: Global stabilization of three-dimensional flexible marine risers by boundary control. Ocean Syst. Eng. 1(2), 171–194 (2011)MathSciNetCrossRef Do, K.D.: Global stabilization of three-dimensional flexible marine risers by boundary control. Ocean Syst. Eng. 1(2), 171–194 (2011)MathSciNetCrossRef
29.
Zurück zum Zitat He, W., Ge, S.S., How, B.V., Choo, Y.S., Hong, K.S.: Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011)MathSciNetCrossRefMATH He, W., Ge, S.S., How, B.V., Choo, Y.S., Hong, K.S.: Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of marine risers with bending couplings. In: Proceedings of the 12th International Conference on Control, Automation and Systems, pp. 1615–1620 (2012) Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of marine risers with bending couplings. In: Proceedings of the 12th International Conference on Control, Automation and Systems, pp. 1615–1620 (2012)
31.
Zurück zum Zitat Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of coupled nonlinear three dimensional marine risers. J. Mar. Sci. Appl. 12(1), 72–88 (2013)CrossRef Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of coupled nonlinear three dimensional marine risers. J. Mar. Sci. Appl. 12(1), 72–88 (2013)CrossRef
32.
Zurück zum Zitat Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of two-dimensional marine risers with bending couplings. J. Sound Vib. 332(16), 3605–3622 (2013)CrossRef Nguyen, T.L., Do, K.D., Pan, J.: Boundary control of two-dimensional marine risers with bending couplings. J. Sound Vib. 332(16), 3605–3622 (2013)CrossRef
33.
Zurück zum Zitat Krstic, M., Siranosian, A.A., Balogh, A., Guo, B.-Z.: Control of strings and flexible beams by backstepping boundary control. American Control Conference, New York, pp. 882–887 (2007) Krstic, M., Siranosian, A.A., Balogh, A., Guo, B.-Z.: Control of strings and flexible beams by backstepping boundary control. American Control Conference, New York, pp. 882–887 (2007)
34.
Zurück zum Zitat Krstic, M., Siranosian, A.A., Smyshlyaev, A.: Backstepping boundary controllers and observers for the slender timoshenko beam: Part idesign. In: American Control Conference, Minnesota, pp. 2412–2417 (2006) Krstic, M., Siranosian, A.A., Smyshlyaev, A.: Backstepping boundary controllers and observers for the slender timoshenko beam: Part idesign. In: American Control Conference, Minnesota, pp. 2412–2417 (2006)
35.
Zurück zum Zitat Krstic, M., Siranosian, A.A., Smyshlyaev, A., Bement, M.: Backstepping boundary controllers and observers for the slender timoshenko beam: Part ii-stability and simulations. In: Proceedings of the 45th IEEE Conference on Decision & Control, pp. 3938–3943 (2006) Krstic, M., Siranosian, A.A., Smyshlyaev, A., Bement, M.: Backstepping boundary controllers and observers for the slender timoshenko beam: Part ii-stability and simulations. In: Proceedings of the 45th IEEE Conference on Decision & Control, pp. 3938–3943 (2006)
36.
37.
Zurück zum Zitat Evans, L.: Partial Differential Equations. American Mathematical Society, Providence (2000) Evans, L.: Partial Differential Equations. American Mathematical Society, Providence (2000)
38.
Zurück zum Zitat Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Oxford (2003)MATH Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Oxford (2003)MATH
39.
Zurück zum Zitat Richtmyer, R.D., Morton, K.W.: Difference Methods for Initial-Value Problems. Wiley, New York (1967)MATH Richtmyer, R.D., Morton, K.W.: Difference Methods for Initial-Value Problems. Wiley, New York (1967)MATH
40.
Zurück zum Zitat Do, K.D.: Hamilton-Jacobi equation for optimal control of nonlinear stochastic distributed parameter systems applied to air pollution process. Appl. Math. Sci. 8(57), 2801–2816 (2014)MathSciNet Do, K.D.: Hamilton-Jacobi equation for optimal control of nonlinear stochastic distributed parameter systems applied to air pollution process. Appl. Math. Sci. 8(57), 2801–2816 (2014)MathSciNet
41.
42.
Zurück zum Zitat Do, K.D.: Global Path-Following Control of Underactuated Ships Under Deterministic and stochastic sea Loads. Robotica, In Press, Cambridge (2015) Do, K.D.: Global Path-Following Control of Underactuated Ships Under Deterministic and stochastic sea Loads. Robotica, In Press, Cambridge (2015)
Metadaten
Titel
Boundary control of nonlinear elastic systems
verfasst von
K. D. Do
Publikationsdatum
01.06.2016
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2016
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-015-0907-5

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