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August  2018, 38(8): 3789-3802. doi: 10.3934/dcds.2018164

Impulsive control of conservative periodic equations and systems: Variational approach

Pavel Drábek 1,, and  Martina Langerová 2,

1. 

Department of Mathematics and NTIS, University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic
2. 

NTIS, University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic

* Corresponding author: Pavel Drábek

Received  July 2017 Revised  November 2017 Published  May 2018

Using the variational structure of the second order periodic problems we find an optimal impulsive control which forces the conservative system into a periodic motion. In particular, our main results concern the system of charged planar pendulums with external disturbances and neglected friction. Such a system might serve as a model for coupled micromechanical array.

Keywords: Periodic problems of the second order, impulsive control, pendulum equation, micromechanical array, variational methods.
Mathematics Subject Classification: 34A37, 34C15, 34C25, 49J15.
Citation: Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164
References:
[1]

N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976. Google Scholar

[2]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Harlow, Longman, 1993.  Google Scholar

[3]

E. Buks and M. L. Roukes, Electrically tunable collective response in a coupled micromechanical array, J. Microelectromech. Syst., 11 (2002), 802-807.  doi: 10.1109/JMEMS.2002.805056.  Google Scholar

[4]

T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297.  doi: 10.1007/BF00940627.  Google Scholar

[5]

T. E. Carter, Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. Control, 10 (2000), 219-227.  doi: 10.1023/A:1008376427023.  Google Scholar

[6]

P. Drábek and M. Langerová, Quasilinear boundary value problem with impulses: Variational approach to resonance problem, Bound. Value Probl., 2014 (2014), 14pp. doi: 10.1186/1687-2770-2014-64.  Google Scholar

[7]

P. Drábek and M. Langerová, On the second order periodic problem at resonance with impulses, J. Math. Anal. Appl., 428 (2015), 1339-1353.  doi: 10.1016/j.jmaa.2015.03.075.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, 2nd edition, Birkhäuser, Springer Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[9]

B. S. Kalinin, On oscillations of mathematical pendulum with striking impulse, Differ. Uravn., 7 (1969), 1267-1274.   Google Scholar

[10]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Cambridge, 1989. doi: 10.1142/0906.  Google Scholar

[11]

X. Liu and A. R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. Probl. Eng., 2 (1996), 277-299.  doi: 10.1155/S1024123X9600035X.  Google Scholar

[12]

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[13]

J. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, in Nonlinear Analysis and Optimization, Lect. Notes Math., (1984), 181-192.  doi: 10.1007/BFb0101500.  Google Scholar

[14]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[15]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022.  Google Scholar

[16]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[17]

Y. Tian and W. Ge, Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 51 (2008), 509-527.  doi: 10.1017/S0013091506001532.  Google Scholar

[18]

M. Willem, Oscillations forcées de systémes hamiltoniens, Publications Mathematiques de la Faculté des Sciences de Besancon, Besancon, 1981. Google Scholar

[19]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272, Springer-Verlag Berlin Heidelberg, 2001.  Google Scholar

show all references

References:
[1]

N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976. Google Scholar

[2]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Harlow, Longman, 1993.  Google Scholar

[3]

E. Buks and M. L. Roukes, Electrically tunable collective response in a coupled micromechanical array, J. Microelectromech. Syst., 11 (2002), 802-807.  doi: 10.1109/JMEMS.2002.805056.  Google Scholar

[4]

T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297.  doi: 10.1007/BF00940627.  Google Scholar

[5]

T. E. Carter, Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dynam. Control, 10 (2000), 219-227.  doi: 10.1023/A:1008376427023.  Google Scholar

[6]

P. Drábek and M. Langerová, Quasilinear boundary value problem with impulses: Variational approach to resonance problem, Bound. Value Probl., 2014 (2014), 14pp. doi: 10.1186/1687-2770-2014-64.  Google Scholar

[7]

P. Drábek and M. Langerová, On the second order periodic problem at resonance with impulses, J. Math. Anal. Appl., 428 (2015), 1339-1353.  doi: 10.1016/j.jmaa.2015.03.075.  Google Scholar

[8]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, 2nd edition, Birkhäuser, Springer Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[9]

B. S. Kalinin, On oscillations of mathematical pendulum with striking impulse, Differ. Uravn., 7 (1969), 1267-1274.   Google Scholar

[10]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Cambridge, 1989. doi: 10.1142/0906.  Google Scholar

[11]

X. Liu and A. R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. Probl. Eng., 2 (1996), 277-299.  doi: 10.1155/S1024123X9600035X.  Google Scholar

[12]

J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.  doi: 10.1016/0022-0396(84)90180-3.  Google Scholar

[13]

J. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, in Nonlinear Analysis and Optimization, Lect. Notes Math., (1984), 181-192.  doi: 10.1007/BFb0101500.  Google Scholar

[14]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[15]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022.  Google Scholar

[16]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[17]

Y. Tian and W. Ge, Applications of variational methods to boundary-value problem for impulsive differential equations, Proc. Edinb. Math. Soc., 51 (2008), 509-527.  doi: 10.1017/S0013091506001532.  Google Scholar

[18]

M. Willem, Oscillations forcées de systémes hamiltoniens, Publications Mathematiques de la Faculté des Sciences de Besancon, Besancon, 1981. Google Scholar

[19]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272, Springer-Verlag Berlin Heidelberg, 2001.  Google Scholar

Figure 1.  A model of 2 coupled charged pendulums.
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Figure 2.  A model of $N$ mutually attracted pendulums.
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