On the Existence of Periodic Solutions with Applications to Extremum-Seeking

This paper provides two results that are useful in the study of the existence and the stability properties of a periodic solution for a given dynamical system. The first result deals with scalar time-periodic systems and establishes the equivalence of the existence of a periodic solution and the existence of a bounded solution. The second result provides sufficient conditions for the existence and the stability of a periodic solution for a time-periodic dynamical system. Both results are applied to extremum seeking problems for a static output map with no plant dynamics and novel non-local results are provided without the use of averaging theorems and singular perturbation arguments.