Topological Methods in Nonlinear Analysis

On stability and controllability for semigroup actions

Josiney A. Souza, Hélio V.M. Tozatti, and Victor H.L. Rocha

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This paper deals with stability and controllability for semigroup actions by using the topological method of admissible family of open coverings. The main results state a relationship of stable sets and control sets. The classical notion of controllability relates to the Poisson stability. The concept of prolongational control set relates to the Lyapunov stability.

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Topol. Methods Nonlinear Anal., Volume 48, Number 1 (2016), 1-29.

First available in Project Euclid: 30 September 2016

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Souza, Josiney A.; Tozatti, Hélio V.M.; Rocha, Victor H.L. On stability and controllability for semigroup actions. Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 1--29. doi:10.12775/TMNA.2016.039.

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  • N.P. Bhatia and O. Hajek, Local semi-dynamical systems, Lecture Notes in Mathematics 90, Springer–Verlag, 1969.
  • N.P. Bhatia and G.P. Szegő, Dynamical systems: stability theory and applications, Lecture Notes in Mathematics 35, Springer–Verlag, 1967.
  • ––––, Stability theory of Dynamical Systems, Springer–Verlag, 1970.
  • C.J. Braga Barros and L.A.B. San Martin, On the action of semigroups in fiber bundles, Mat. Contemp. 13 (1997), 1–19.
  • C.J. Braga Barros and J.A. Souza, Attractors and chain recurrence for semigroup actions, J. Dynam. Differential Equations 22 (2010), 723–740.
  • C.J. Braga Barros, V.H.L. Rocha and J.A. Souza, Lyapunov stability for semigroup actions, Semigroup Forum 88 (2014), 227–249.
  • M. Patrão L.A.B. San Martin, Morse decomposition of semiflows on fiber bundles, Discrete Contin. Dyn. Syst. 17 (2007), 113–139.
  • ––––, Semiflows on topological spaces: chain transitivity and semigroups, J. Dynam. Diferential Equations 19 (2007), 155–180.
  • S.A. Raminelli and J.A. Souza, Global attractors for semigroup actions, J. Math. Anal. Appl. 407 (2013), 316–327.
  • L.A.B. San Martin, Invariant control sets on flag manifolds, Math. Control Signals Systems 6 (1993), 41–61.
  • ––––, Control sets and semigroups in semi-simple Lie groups, Semigroups in Algebra, Geometry and Analysis, Gruyter Verlag, 1994.
  • L.A.B. San Martin and P.A. Tonelli, Semigroup actions on homogeneous spaces, Semigroup Forum 50 (1995), 59–88.
  • J.A. Souza, Complete Lyapunov functions of control systems, Systems Control Lett. 61 (2012), 322–326.
  • ––––, Lebesgue covering lemma on nonmetric spaces, Internat. J. Math. 24 (2013), 1350018, 1–12.
  • ––––, On limit behavior of semigroup actions on noncompact space, Proc. Amer. Math. Soc. 140 (2012), 3959–3972.
  • ––––, On limit behavior of skew-product transformation semigroups, Math. Nachr. 287 (2014), 91–104.
  • ––––, Recurrence theorem for semigroup actions, Semigroup Forum 83 (2011), 351–370.
  • J.A. Souza and H.V.M. Tozatti, Prolongational limit sets of control systems, J. Differential Equations 254 (2013), 2183–2195.
  • ––––, Some aspect of stability for semigroup actions, J. Dynam. Differential Equations 26 (2014), 631–654.