Inverse variational principles for control systems

Abstract

Abstract Inspired by Katok’s idea in defining measure-theoretic entropy, we introduce several measure-theoretic invariance entropies in control systems via spanning sets. Properties of them are investigated. Especially, we show that the measure-theoretic invariance entropies of a given set can be characterized by its relatively invariant subsets. More importantly, we obtain two inverse variational principles connecting measure-theoretic invariance entropy and topological invariance entropy, which help us to establish two formulae for the measure-theoretic invariance entropies of control sets for linear control systems, including linear continuous-time and discrete-time systems.