| Contents |
| The talk is devoted to the existence of solutions to the following
single-valued or multivalued differential problem $\dot x(t)=f(x(t))$
[resp. $\in F(x(t))$] for a.e. $t\geq 0$, $x(t^+):=\lim_{s\to t^+}
x(s)\in I(x(t))$ for $x(t)\in M\subset
\partial K$, $x(t)\in K$ for every $t\geq 0$, and
$x(0)=x(T)$ for some $T>0$,
where $K\subset\mathbb R^n$ is a closed set and $I:M\to 2^{\mathbb R^n}$ is
an impulse function. Every solution to this problem is said to be viable periodic.
Without tangency conditions on the whole set $K$, we allow trajectories for a
non-impulsive problem, even all of them (then we say that $K$ is totally
leaky), to leave $K$ through a so-called exit set. To prevent this, we
place in an exit set a barrier $M$, and define an impulse function $I$
possibly moving some trajectories back to the set $K$.
We will look for topological sufficient conditions for the existence of viable
periodic trajectories for an impulsive system. An `impulsive index' will
be provided as a suitable homotopy invariant, and its properties will be
presented. It will be defined as a fixed point index of a corresponding
multivalued (or single-valued) map on the exit set induced by a flow and an
impulse function.
The research is a continuation and development of the one presented in Gabor
G.:
The existence of viable trajectories in state-dependent impulsive
systems. Nonlinear Anal. 72 (2010), 3828-3836. |
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