doi: 10.3934/mcrf.2021042
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Optimal control of transverse vibration of a moving string with time-varying lengths

School of Mathematics and Statistics & Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: Bing Sun

Received  August 2019 Revised  September 2020 Early access September 2021

Fund Project: The author is supported by the National Natural Science Foundation of China grant 11471036.
This article was recruited by Sergey Dashkovskiy

In this article, we are concerned with optimal control for the transverse vibration of a moving string with time-varying lengths. In the fixed final time horizon case, the Pontryagin maximum principle is established for the investigational system with a moving boundary, owing to the Dubovitskii and Milyutin functional analytical approach. A remark then follows for discussing the utilization of obtained necessary optimality condition.

Citation: Bing Sun. Optimal control of transverse vibration of a moving string with time-varying lengths. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021042
References:
[1]

F. R. Archibald and A. G. Emslie, The vibration of a string having uniform motion along its length, ASME J. Appl. Mech., 25 (1958), 347-348.   Google Scholar

[2]

L. Cai, Active vibration control of axially moving continua, in Proceedings Volume 2620, International Conference on Intelligent Manufacturing (eds. S. Yang, J. Zhou and C.-G. Li), Society of Photo-optical Instrumentation Engineers (SPIE), (1995), 780–785. Google Scholar

[3]

W.-L. Chan and B.-Z. Guo, Optimal birth control of population dynamics, J. Math. Anal. Appl., 144 (1989), 532-552.  doi: 10.1016/0022-247X(89)90350-8.  Google Scholar

[4]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. I, Springer-Verlag, Berlin, 1968.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Wiley-Interscience, New York, 1988.  Google Scholar

[6]

H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, 62, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.  Google Scholar

[7]

H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005.  Google Scholar

[8]

J. A. Gibson and J. F. Lowinger, A predictive min-H method to improve convergence to optimal solutions, Internat. J. Control, 19 (1974), 575-592.  doi: 10.1080/00207177408932654.  Google Scholar

[9]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 67, Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-80684-1.  Google Scholar

[10]

M. Gugat, Optimal energy control in finite time by varying the length of the string, SIAM J. Control Optim., 46 (2007), 1705-1725.  doi: 10.1137/06065636x.  Google Scholar

[11]

M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA J. Math. Control Inform., 25 (2008), 111-121.  doi: 10.1093/imamci/dnm014.  Google Scholar

[12]

B.-Z. Guo, Asymptotic behavior of the energy of vibration of a moving string with varying lengths, J. Vib. Control, 6 (2000), 491-507.  doi: 10.1177/107754630000600401.  Google Scholar

[13]

B.-Z. Guo and B. Sun, A new algorithm for finding numerical solutions of optimal feedback control, IMA J. Math. Control Inform., 26 (2009), 95-104.  doi: 10.1093/imamci/dnn001.  Google Scholar

[14]

B.-Z. Guo and J.-X. Wang, The unbounded energy solution for free vibration of an axially moving string, J. Vib. Control, 6 (2000), 651-665.  doi: 10.1177/107754630000600501.  Google Scholar

[15]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Systems & Control: Foundations & Applications, Birhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[16]

W. L. Miranker, The wave equation in a medium in motion, IBM J. Res. Develop., 4 (1960), 36-42.  doi: 10.1147/rd.41.0036.  Google Scholar

[17]

M. PakdemirliA. G. Ulsoy and A. Ceranoglu, Transverse vibration of an axially accelerating string, J. Sound Vibration, 169 (1994), 179-196.  doi: 10.1006/jsvi.1994.1012.  Google Scholar

[18]

V. V. Popov, Vibrations of a segment of a variable-length longitudinally-moving string, J. Appl. Math. Mech., 50 (1986), 161-164.  doi: 10.1016/0021-8928(86)90100-0.  Google Scholar

[19]

B. Sun and B.-Z. Guo, Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control, IEEE Trans. Automat. Control, 60 (2015), 3012-3017.  doi: 10.1109/TAC.2015.2406976.  Google Scholar

[20]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, in Mathematics Applied to Engineering, Modelling, and Social Issues (eds. F. T. Smith, H. Dutta and J. N. Mordeson), Studies in Systems, Decision and Control, 200, Springer, Cham, (2019), 363–420. doi: 10.1007/978-3-030-12232-4_12.  Google Scholar

[21]

B. Sun and M.-X. Wu, Optimal control of a continuum model for a highly re-entrant manufacturing system, Trans. Inst. Meas. Control, 41 (2019), 1373-1382.  doi: 10.1177/0142331218778100.  Google Scholar

[22]

I. B. Vapnyarskii, Optimality, Sufficient Conditions for, Report of Hong Kong SARS Expert Committee, Encyclopedia of Mathematics, 2020. Available from: http://www.encyclopediaofmath.org/index.php?title=Optimality,_sufficient_conditions_for&oldid=13932. Google Scholar

[23]

J. A. Wickert and C. D. Mote Jr, On the energetics of axially moving continua, J. Acoust. Soc. Am., 85 (1989), 1365-1368.  doi: 10.1121/1.397418.  Google Scholar

[24] J.-X. XingC.-R. Zhang and H.-Z. Xu, Basics of Optimal Control Application (Chinese), Series of Textbooks for Graduate Students in Control Science and Engineering, Science Press, Beijing, 2003.   Google Scholar
[25]

K.-J. YangK.-S. Hong and F. Matsuno, Robust boundary control of an axially moving string by using a PR transfer function, IEEE Trans. Automat. Control, 50 (2005), 2053-2058.  doi: 10.1109/TAC.2005.860252.  Google Scholar

[26]

S. ZhangW. He and D. Huang, Active vibration control for a flexible string system with input backlash, IET Control Theory Appl., 10 (2016), 800-805.  doi: 10.1049/iet-cta.2015.1044.  Google Scholar

[27]

Z. ZhaoY. LiuF. Guo and Y. Fu, Vibration control and boundary tension constraint of an axially moving string system, Nonlinear Dynam., 89 (2017), 2431-2440.  doi: 10.1007/s11071-017-3595-x.  Google Scholar

[28]

W.-D. Zhu and B.-Z. Guo, Free and forced vibration of an axially moving string with an arbitrary velocity profile, ASME J. Appl. Mech., 65 (1998), 901-907.  doi: 10.1115/1.2791932.  Google Scholar

show all references

References:
[1]

F. R. Archibald and A. G. Emslie, The vibration of a string having uniform motion along its length, ASME J. Appl. Mech., 25 (1958), 347-348.   Google Scholar

[2]

L. Cai, Active vibration control of axially moving continua, in Proceedings Volume 2620, International Conference on Intelligent Manufacturing (eds. S. Yang, J. Zhou and C.-G. Li), Society of Photo-optical Instrumentation Engineers (SPIE), (1995), 780–785. Google Scholar

[3]

W.-L. Chan and B.-Z. Guo, Optimal birth control of population dynamics, J. Math. Anal. Appl., 144 (1989), 532-552.  doi: 10.1016/0022-247X(89)90350-8.  Google Scholar

[4]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. I, Springer-Verlag, Berlin, 1968.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Wiley-Interscience, New York, 1988.  Google Scholar

[6]

H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, 62, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795.  Google Scholar

[7]

H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005.  Google Scholar

[8]

J. A. Gibson and J. F. Lowinger, A predictive min-H method to improve convergence to optimal solutions, Internat. J. Control, 19 (1974), 575-592.  doi: 10.1080/00207177408932654.  Google Scholar

[9]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, 67, Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-80684-1.  Google Scholar

[10]

M. Gugat, Optimal energy control in finite time by varying the length of the string, SIAM J. Control Optim., 46 (2007), 1705-1725.  doi: 10.1137/06065636x.  Google Scholar

[11]

M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA J. Math. Control Inform., 25 (2008), 111-121.  doi: 10.1093/imamci/dnm014.  Google Scholar

[12]

B.-Z. Guo, Asymptotic behavior of the energy of vibration of a moving string with varying lengths, J. Vib. Control, 6 (2000), 491-507.  doi: 10.1177/107754630000600401.  Google Scholar

[13]

B.-Z. Guo and B. Sun, A new algorithm for finding numerical solutions of optimal feedback control, IMA J. Math. Control Inform., 26 (2009), 95-104.  doi: 10.1093/imamci/dnn001.  Google Scholar

[14]

B.-Z. Guo and J.-X. Wang, The unbounded energy solution for free vibration of an axially moving string, J. Vib. Control, 6 (2000), 651-665.  doi: 10.1177/107754630000600501.  Google Scholar

[15]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Systems & Control: Foundations & Applications, Birhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[16]

W. L. Miranker, The wave equation in a medium in motion, IBM J. Res. Develop., 4 (1960), 36-42.  doi: 10.1147/rd.41.0036.  Google Scholar

[17]

M. PakdemirliA. G. Ulsoy and A. Ceranoglu, Transverse vibration of an axially accelerating string, J. Sound Vibration, 169 (1994), 179-196.  doi: 10.1006/jsvi.1994.1012.  Google Scholar

[18]

V. V. Popov, Vibrations of a segment of a variable-length longitudinally-moving string, J. Appl. Math. Mech., 50 (1986), 161-164.  doi: 10.1016/0021-8928(86)90100-0.  Google Scholar

[19]

B. Sun and B.-Z. Guo, Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control, IEEE Trans. Automat. Control, 60 (2015), 3012-3017.  doi: 10.1109/TAC.2015.2406976.  Google Scholar

[20]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, in Mathematics Applied to Engineering, Modelling, and Social Issues (eds. F. T. Smith, H. Dutta and J. N. Mordeson), Studies in Systems, Decision and Control, 200, Springer, Cham, (2019), 363–420. doi: 10.1007/978-3-030-12232-4_12.  Google Scholar

[21]

B. Sun and M.-X. Wu, Optimal control of a continuum model for a highly re-entrant manufacturing system, Trans. Inst. Meas. Control, 41 (2019), 1373-1382.  doi: 10.1177/0142331218778100.  Google Scholar

[22]

I. B. Vapnyarskii, Optimality, Sufficient Conditions for, Report of Hong Kong SARS Expert Committee, Encyclopedia of Mathematics, 2020. Available from: http://www.encyclopediaofmath.org/index.php?title=Optimality,_sufficient_conditions_for&oldid=13932. Google Scholar

[23]

J. A. Wickert and C. D. Mote Jr, On the energetics of axially moving continua, J. Acoust. Soc. Am., 85 (1989), 1365-1368.  doi: 10.1121/1.397418.  Google Scholar

[24] J.-X. XingC.-R. Zhang and H.-Z. Xu, Basics of Optimal Control Application (Chinese), Series of Textbooks for Graduate Students in Control Science and Engineering, Science Press, Beijing, 2003.   Google Scholar
[25]

K.-J. YangK.-S. Hong and F. Matsuno, Robust boundary control of an axially moving string by using a PR transfer function, IEEE Trans. Automat. Control, 50 (2005), 2053-2058.  doi: 10.1109/TAC.2005.860252.  Google Scholar

[26]

S. ZhangW. He and D. Huang, Active vibration control for a flexible string system with input backlash, IET Control Theory Appl., 10 (2016), 800-805.  doi: 10.1049/iet-cta.2015.1044.  Google Scholar

[27]

Z. ZhaoY. LiuF. Guo and Y. Fu, Vibration control and boundary tension constraint of an axially moving string system, Nonlinear Dynam., 89 (2017), 2431-2440.  doi: 10.1007/s11071-017-3595-x.  Google Scholar

[28]

W.-D. Zhu and B.-Z. Guo, Free and forced vibration of an axially moving string with an arbitrary velocity profile, ASME J. Appl. Mech., 65 (1998), 901-907.  doi: 10.1115/1.2791932.  Google Scholar

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