A measure-theoretic proof of the Markov property for hybrid systems with Markovian inputs

The behavior of a general hybrid system in discrete time can be represented by a non-linear difference equation x(k + 1) = F_k(x(k), thetas(k)), where thetas(k) is assumed to be a finite state Markov chain. An important step in the stability analysis of these systems is to establish the Markov property of (x(k), thetas(k)). There are, however, no complete proofs of this property which are simple to understand. This paper aims to correct this problem by presenting a complete and explicit proof, which uses only basic measure-theoretical concepts