On the state of behaviors

The theme of the present paper is the study of the concept of state and the corresponding state maps in the context of Willems’ behavioral theory. We concentrate on Markovian system and their representation in terms of first order difference or differential systems. We follow by a full analysis of the special case of state systems, the embedding of a linear system in a state system via the use of state maps arriving at state representations or, equivalently, to a realization theory for behaviors. Minimality is defined and characterized and a state space isomorphism theorem is established. Realization procedures based on the shift realization are developed as well as a rigorous analysis of the construction of state maps. The paper owes much to Rapisarda and Willems [P. Rapisarda, J.C. Willems, State maps for linear systems, SIAM J. Contr. Optim. 35 (1997) 1053–1091].