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On a distributed control problem for a coupled chemotaxisfluid model
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, 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, 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, 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, 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, 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, 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, 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, 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. 
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