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 and Continuous Dynamical Systems, 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.

show all references

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.

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

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[2]

João Fialho, Feliz Minhós. High order periodic impulsive problems. Conference Publications, 2015, 2015 (special) : 446-454. doi: 10.3934/proc.2015.0446

[3]

Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001

[4]

Masayuki Sato, Naoki Fujita, A. J. Sievers. Logic operations demonstrated with localized vibrations in a micromechanical cantilever array. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1287-1298. doi: 10.3934/dcdss.2011.4.1287

[5]

Christos Gavriel, Richard Vinter. Regularity of minimizers for second order variational problems in one independent variable. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 547-557. doi: 10.3934/dcds.2011.29.547

[6]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[7]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial and Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[8]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

[9]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177

[10]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851

[11]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[12]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070

[13]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations and Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051

[14]

J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431

[15]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445

[16]

Doria Affane, Mustapha Fateh Yarou. Well-posed control problems related to second-order differential inclusions. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021042

[17]

Soumia Saïdi, Fatima Fennour. Second-order problems involving time-dependent subdifferential operators and application to control. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022019

[18]

Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451

[19]

Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457

[20]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (315)
  • HTML views (186)
  • Cited by (0)

Other articles
by authors

[Back to Top]