
Previous Article
On a distributed control problem for a coupled chemotaxisfluid model
 DCDSB Home
 This Issue

Next Article
Asymptotic behaviour of the solutions to a virus dynamics model with diffusion
Restrictions to the use of timedelayed feedback control in symmetric settings
Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA 
We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the noninvasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant twodimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts noninvasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled StewartLandau oscillators and a system of two coupled lasers.
References:
[1] 
G. Brown, C. Postlethwaite and M. Silber, Timedelayed feedback control of unstable periodic orbits near a subcritical hopf bifurcation, Physica D: Nonlinear Phenomena, 240 (2011), 859871. doi: 10.1016/j.physd.2010.12.011. 
[2] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using ddebiftool, ACM Trans. Math. Softw., 28 (2002), 121. doi: 10.1145/513001.513002. 
[3] 
K. Engelborghs, T. Luzyanina and G. Samaey, DDEBIFTOOL v. 2. 00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Department of Computer Science, Technical report, KU Leuven, Technical Report TW330, Leuven, Belgium, 2001. 
[4] 
B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319341. doi: 10.1098/rsta.2009.0232. 
[5] 
B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to alloptical control of a semiconductor laser, Physical Review E, 77 (2008), 066207, 9pp doi: 10.1103/PhysRevE.77.066207. 
[6] 
E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive Stabilization of Periodic Orbits in O_{4}Symmetrically Coupled Systems Near a Hopf Bifurcation Point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18 doi: 10.1142/S0218127417500870. 
[7] 
E. Hooton and A. Amann, Analytical limitation for timedelayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101. 
[8] 
H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207210. 
[9] 
C. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467, 20pp. doi: 10.1098/rsta.2012.0467. 
[10] 
K. Pyragas, Continuous control of chaos by selfcontrolling feedback, Physics Letters A, 170 (1992), 421428. 
[11] 
S. Schikora, P. Hövel, H. J. Wünsche, E. Schöll and F. Henneberger, Alloptical noninvasive control of unstable steady states in a semiconductor laser, Physical Review Letters, 97(2006), 213902. 
[12] 
I. Schneider, Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472. 
[13] 
I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014. 
[14] 
I. Schneider and M. Bosewitz, Eliminating restrictions of timedelayed feedback control using equivariance, Discrete and Continuous Dynamical Systems Series A, 36 (2016), 451467. doi: 10.3934/dcds.2016.36.451. 
[15] 
I. Schneider and B. Fiedler, Symmetrybreaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. 
[16] 
J. Sieber, Generic stabilizability for timedelayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593. 
[17] 
J. Sieber, A. GonzalezBuelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101. 
[18] 
M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904. 
[19] 
S. Yanchuk, K. R. Schneider and L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Physical Review E, 69 (2004), 056221, URL http://link.aps.org/doi/10.1103/PhysRevE.69.056221. 
show all references
References:
[1] 
G. Brown, C. Postlethwaite and M. Silber, Timedelayed feedback control of unstable periodic orbits near a subcritical hopf bifurcation, Physica D: Nonlinear Phenomena, 240 (2011), 859871. doi: 10.1016/j.physd.2010.12.011. 
[2] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using ddebiftool, ACM Trans. Math. Softw., 28 (2002), 121. doi: 10.1145/513001.513002. 
[3] 
K. Engelborghs, T. Luzyanina and G. Samaey, DDEBIFTOOL v. 2. 00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Department of Computer Science, Technical report, KU Leuven, Technical Report TW330, Leuven, Belgium, 2001. 
[4] 
B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319341. doi: 10.1098/rsta.2009.0232. 
[5] 
B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to alloptical control of a semiconductor laser, Physical Review E, 77 (2008), 066207, 9pp doi: 10.1103/PhysRevE.77.066207. 
[6] 
E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive Stabilization of Periodic Orbits in O_{4}Symmetrically Coupled Systems Near a Hopf Bifurcation Point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18 doi: 10.1142/S0218127417500870. 
[7] 
E. Hooton and A. Amann, Analytical limitation for timedelayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101. 
[8] 
H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207210. 
[9] 
C. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467, 20pp. doi: 10.1098/rsta.2012.0467. 
[10] 
K. Pyragas, Continuous control of chaos by selfcontrolling feedback, Physics Letters A, 170 (1992), 421428. 
[11] 
S. Schikora, P. Hövel, H. J. Wünsche, E. Schöll and F. Henneberger, Alloptical noninvasive control of unstable steady states in a semiconductor laser, Physical Review Letters, 97(2006), 213902. 
[12] 
I. Schneider, Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472. 
[13] 
I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014. 
[14] 
I. Schneider and M. Bosewitz, Eliminating restrictions of timedelayed feedback control using equivariance, Discrete and Continuous Dynamical Systems Series A, 36 (2016), 451467. doi: 10.3934/dcds.2016.36.451. 
[15] 
I. Schneider and B. Fiedler, Symmetrybreaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. 
[16] 
J. Sieber, Generic stabilizability for timedelayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593. 
[17] 
J. Sieber, A. GonzalezBuelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101. 
[18] 
M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904. 
[19] 
S. Yanchuk, K. R. Schneider and L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Physical Review E, 69 (2004), 056221, URL http://link.aps.org/doi/10.1103/PhysRevE.69.056221. 
[1] 
Isabelle Schneider, Matthias Bosewitz. Eliminating restrictions of timedelayed feedback control using equivariance. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 451467. doi: 10.3934/dcds.2016.36.451 
[2] 
Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629647. doi: 10.3934/mbe.2009.6.629 
[3] 
Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 3439. 
[4] 
Ran Dong, Xuerong Mao. Asymptotic stabilization of continuoustime periodic stochastic systems by feedback control based on periodic discretetime observations. Mathematical Control and Related Fields, 2020, 10 (4) : 715734. doi: 10.3934/mcrf.2020017 
[5] 
Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete and Continuous Dynamical Systems  S, 2019, 12 (7) : 20192034. doi: 10.3934/dcdss.2019130 
[6] 
Ta T.H. Trang, Vu N. Phat, Adly Samir. Finitetime stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303315. doi: 10.3934/jimo.2016.12.303 
[7] 
Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $expectation framework. Discrete and Continuous Dynamical Systems  B, 2022, 27 (2) : 883901. doi: 10.3934/dcdsb.2021072 
[8] 
Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete and Continuous Dynamical Systems  S, 2008, 1 (2) : 197205. doi: 10.3934/dcdss.2008.1.197 
[9] 
Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete and Continuous Dynamical Systems  B, 2011, 15 (3) : 893914. doi: 10.3934/dcdsb.2011.15.893 
[10] 
Martin Gugat, Markus Dick. Timedelayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control and Related Fields, 2011, 1 (4) : 469491. doi: 10.3934/mcrf.2011.1.469 
[11] 
Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete and Continuous Dynamical Systems  B, 2020, 25 (4) : 14391467. doi: 10.3934/dcdsb.2019235 
[12] 
Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems  S, 2018, 11 (6) : 11031119. doi: 10.3934/dcdss.2018063 
[13] 
Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems  B, 2017, 22 (6) : 20672088. doi: 10.3934/dcdsb.2017085 
[14] 
John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 57. 
[15] 
Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 629652. doi: 10.3934/dcds.2007.17.629 
[16] 
Shu Zhang, Jian Xu. Timevarying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems  B, 2011, 16 (2) : 653668. doi: 10.3934/dcdsb.2011.16.653 
[17] 
Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 23692387. doi: 10.3934/dcds.2013.33.2369 
[18] 
Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed nonaxially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete and Continuous Dynamical Systems  B, 2018, 23 (6) : 22992337. doi: 10.3934/dcdsb.2018101 
[19] 
Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109124. doi: 10.3934/jgm.2015.7.109 
[20] 
Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. Journal of Geometric Mechanics, 2020, 12 (3) : 395420. doi: 10.3934/jgm.2020022 
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]