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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 and 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. 

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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.

[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.

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. 

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 
[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.

[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.

[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.

[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.

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