The strong stability of the Perron–Frobenius semigroup and almost global attractivity

We discuss some useful properties of the solution map (flow) of a nonlinear dynamical system with a finite-dimensional state space. Then, we introduce the Perron–Frobenius semigroup, and we prove that it is a positive strongly continuous semigroup of contractions. We show that, given a nonlinear system and an invariant set, this set is an almost global attractor if and only if certain Perron–Frobenius semigroups associated to the nonlinear system are strongly stable. Unlike other works on the Perron–Frobenius semigroup from the literature, we do not require the existence of a compact and invariant state-space for the dynamical system, we allow trajectories with finite escape time, and we do not require the attractor to be locally (Lyapunov) stable. Two simple examples are used throughout the paper to illustrate the theory.