Abstract

Considering a family of discrete-time dynamical systems \(x_{k+1} \in F_p(x_k)\), the author gives sufficient conditions implying reachability (or local reachability) in the sense that, for any \(p\) sufficiently close to \(p_0\), \(\mathcal{R}_p = X\) (or \(0 \in \text{int }\mathcal{R}_p\)) where \(\mathcal{R}_p\) denotes the set of points \(u \in X\) such that there exists at least one trajectory starting from \(p\) and reaching \(u\). These conditions involve lower pseudo-Hausdorff semicontinuity at \(p_0\) of the map \(p \mapsto \text{Graph}(F_p)\). The paper ends with results giving necessary and sufficient conditions for reachability of discrete dynamical systems with closed convex right hand side.