09/09/2016, 15:00 — 15:40 — Amphitheatre Pa2, Mathematics Building
Radoslaw Czaja, University of Silesia at Katowice

Attractors for Dynamical Systems with Impulses

In this talk I will formulate a suitable definition of a global attractor for impulsive dynamical systems, which model the evolution of a continuous process interrupted by abrupt changes of state.

An impulsive dynamical system (IDS) consists of a continuous semigroup $\{\pi(t) : t\geq 0\}$ on a metric space $X$, a nonempty closed subset $M$ of $X$ called an impulsive set, which is “transversal” to the flow of the semigroup, and a continuous function $I : M \to X$ called an impulsive function. Whenever a trajectory for the semigroup $\pi$ hits the set $M$ the impulsive function $I$ redirects it to a new state, defining an impulsive trajectory. Assuming that all impulsive trajectories exist for all times $t\geq 0$, we obtain a possibly discontinuous semigroup $\{\tilde{\pi}(t):t\geq 0\}$.

To describe the long-time behavior of $\tilde{\pi}($ we introduce the notion of a global attractor $\mathcal{A}\subset X$, which is precompact, $\mathcal{A} = \overline{\mathcal{A}}\setminus M$, $\tilde{\pi}$-invariant and $\tilde{\pi}$-attracting all bounded subsets of $X$. Such sets are more suitable for impulsive dynamical systems and better describe their dynamics than compact global attractors known for continuous semigroups.

I will present several properties for this class of precompact global attractors and a theorem on existence of such attractors. The theory has applications to chosen ordinary and partial differential equations with impulsive functions. This is a joint work with E. M. Bonotto, A. N. Carvalho and R. Collegari from the University of São Paulo in São Carlos and M. C. Bortolan from the Federal University of Santa Catarina, Brazil ([1, 2]).

References

1. E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, Journal of Differential Equations, 259 (2015), 2602{2625, doi:10.1016/j.jde.2015.03.033.
2. E. M. Bonotto, M. C. Bortolan, R. Collegari, R. Czaja, Semicontinuity of attractors for impulsive dynamical systems, to appear in Journal of Differential Equations, doi:10.1016/j.jde.2016.06.024.