American Institute of Mathematical Sciences

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

Impulsive control of conservative periodic equations and systems: Variational approach

 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.

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. [2] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Harlow, Longman, 1993. [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. [4] T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297. doi: 10.1007/BF00940627. [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. [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. [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. [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. [9] B. S. Kalinin, On oscillations of mathematical pendulum with striking impulse, Differ. Uravn., 7 (1969), 1267-1274. [10] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Cambridge, 1989. doi: 10.1142/0906. [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. [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. [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. [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. [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. [16] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664. [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. [18] M. Willem, Oscillations forcées de systémes hamiltoniens, Publications Mathematiques de la Faculté des Sciences de Besancon, Besancon, 1981. [19] T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272, Springer-Verlag Berlin Heidelberg, 2001.

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References:
 [1] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976. [2] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Harlow, Longman, 1993. [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. [4] T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297. doi: 10.1007/BF00940627. [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. [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. [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. [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. [9] B. S. Kalinin, On oscillations of mathematical pendulum with striking impulse, Differ. Uravn., 7 (1969), 1267-1274. [10] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Cambridge, 1989. doi: 10.1142/0906. [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. [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. [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. [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. [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. [16] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664. [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. [18] M. Willem, Oscillations forcées de systémes hamiltoniens, Publications Mathematiques de la Faculté des Sciences de Besancon, Besancon, 1981. [19] T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272, Springer-Verlag Berlin Heidelberg, 2001.
A model of 2 coupled charged pendulums.
A model of $N$ mutually attracted pendulums.
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