Mathematical foundations of discrete and functional systems with strong and weak anticipations

Abstract

This paper deals with some mathematical developments to model anticipatory capabilities in discrete and continuous systems. The paper defines weak anticipation and strong anticipation and introduces the concepts of incursive and hyperincursive discrete processes as an extension to recursion. Functional systems represented by differential difference equations with anticipation and/or delay seem to be a very useful tool for describing strong anticipation. Anticipation and delay play a complementary role and synchronization mechanisms seem to be a powerful way to anticipate the evolution of systems with delay. This paper shows finally that the modelling of anticipation in predictive control is the basic mechanism for enhancing the control of the trajectory of systems toward a target.