# American Institute of Mathematical Sciences

July  2011, 16(1): 239-263. doi: 10.3934/dcdsb.2011.16.239

## Stabilization via symmetry switching in hybrid dynamical systems

 1 Chair of Applied Mathematics, University of Paderborn, D-33098 Paderborn, Germany, Germany

Received  June 2010 Revised  February 2011 Published  April 2011

With a view to stabilization issues of hybrid systems exhibiting a regular structure in terms of symmetry, we introduce the concept of symmetry switching and relate symmetry-induced switching strategies to the asymptotic stability of switched linear systems. To this end, a general notion of hybrid symmetries for switched systems is established whereupon orbital switching is treated which builds on the existence of hybrid symmetries. In the main part, we formulate and prove sufficient conditions for asymptotic stability under slow symmetry switching. As an example of both theoretical and practical interest, we examine time-varying networks of dynamical systems and perform stabilization by means of orbital switching. Behind all that, this work is meant to provide the groundwork for the treatment of equivariant bifurcation phenomena of hybrid systems.
Citation: Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239
##### References:
 [1] M. Dellnitz and S. Hage-Packhäuser, A global symmetry framework for Hybrid dynamical systems,, preprint., (). [2] B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: I. Wreath products, Nonlinearity, 9 (1996), 559-574. doi: 10.1088/0951-7715/9/2/016. [3] B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: II. Direct products, Nonlinearity, 9 (1996), 575-599. doi: 10.1088/0951-7715/9/2/017. [4] B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," volume $1309$ of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 1988. [5] M. Golubitsky and I. Stewart, "The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space," volume $200$ of Progress in Mathematics, Birkhäuser, Berlin, 2002. [6] M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol. II," volume $69$ of Applied Mathematical Sciences, Springer, New York, 1988. [7] M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364. doi: 10.1090/S0273-0979-06-01108-6. [8] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. [9] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," volume II of Applied Mathematical Sciences, Cambridge University Press, 1996. [10] J. S. W. Lamb, k-symmetry and return maps of spacetime symmetric flows, Nonlinearity, 11 (1998), 601-629. doi: 10.1088/0951-7715/11/3/011. [11] D. Liberzon, "On New Sufficient Conditions for Stability of Switched Linear Systems," Proceedings of the 2009 European Control Conference, Budapest, Hungary, Aug 2009. [12] D. Liberzon, "Switching in Systems and Control," Systems and Control: Foundations and Applications Birkhäuser, Boston, 2003. [13] J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang and S. Sastry, Dynamical properties of Hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650. [14] J. Lygeros, K. H. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of Hybrid automata, Proceedings of the 38th IEEE Conference on Decision and Control 1999, 3 (1999), 2249-2254. [15] S. N. Simic, K. H. Johansson, J. Lygeros and S. Sastry, Towards a geometric theory of Hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 12 (2005), 649-687. [16] A. van der Schaft and H. Schumacher, "An Introduction to Hybrid Dynamical Systems," Number 251 in Lecture Notes in Control and Information Sciences Springer-Verlag, London, 2000.

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##### References:
 [1] M. Dellnitz and S. Hage-Packhäuser, A global symmetry framework for Hybrid dynamical systems,, preprint., (). [2] B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: I. Wreath products, Nonlinearity, 9 (1996), 559-574. doi: 10.1088/0951-7715/9/2/016. [3] B. Dionne, M. Golubitsky and I. Stewart, Coupled cells with internal symmetry: II. Direct products, Nonlinearity, 9 (1996), 575-599. doi: 10.1088/0951-7715/9/2/017. [4] B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," volume $1309$ of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 1988. [5] M. Golubitsky and I. Stewart, "The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space," volume $200$ of Progress in Mathematics, Birkhäuser, Berlin, 2002. [6] M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol. II," volume $69$ of Applied Mathematical Sciences, Springer, New York, 1988. [7] M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006), 305-364. doi: 10.1090/S0273-0979-06-01108-6. [8] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. [9] P. Holmes, J. L. Lumley and G. Berkooz, "Turbulence, Coherent Structures, Dynamical Systems and Symmetry," volume II of Applied Mathematical Sciences, Cambridge University Press, 1996. [10] J. S. W. Lamb, k-symmetry and return maps of spacetime symmetric flows, Nonlinearity, 11 (1998), 601-629. doi: 10.1088/0951-7715/11/3/011. [11] D. Liberzon, "On New Sufficient Conditions for Stability of Switched Linear Systems," Proceedings of the 2009 European Control Conference, Budapest, Hungary, Aug 2009. [12] D. Liberzon, "Switching in Systems and Control," Systems and Control: Foundations and Applications Birkhäuser, Boston, 2003. [13] J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang and S. Sastry, Dynamical properties of Hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650. [14] J. Lygeros, K. H. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of Hybrid automata, Proceedings of the 38th IEEE Conference on Decision and Control 1999, 3 (1999), 2249-2254. [15] S. N. Simic, K. H. Johansson, J. Lygeros and S. Sastry, Towards a geometric theory of Hybrid systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 12 (2005), 649-687. [16] A. van der Schaft and H. Schumacher, "An Introduction to Hybrid Dynamical Systems," Number 251 in Lecture Notes in Control and Information Sciences Springer-Verlag, London, 2000.
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