Axiomatic foundation of the theory of control systems

Summary The purpose of this paper is to show how an axiomatic development of the theory of control systems can systematize important concepts. Starting with the idea of attainable set (the set of points which can be reached from a given one in a certain time) a set of 6 axioms is given to describe the behaviour of a general control system. Barbashin already gave such an axiomatic approach, and Zubov also used an axiomatic theory for proving stability theorems. But besides the strong form of stability and invariance of a set (given by Zubov) a weak form of these properties can be defined for general control systems. In general, almost every property of the classical results can be translated in a strong and a weak form for control systems. Accordingly, the powerful second method of Liapunov can also be applied in a strong and a weak form, in order to prove strong and weak properties, respectively. In this paper is given the set of basic axioms (more or less equivalent to Barbashin's); some basic results are discussed; the strong and weak form of invariance and stability properties and the corresponding strong and weak Liapunov functions are defined, giving the statement of the fundamental stability theorems (the detailed proofs are given elsewhere).