Jordan algebra and control theory

In order to understand the governing principles of a specific topic, e.g., control design, and to be able to design efficient algorithms a sound mathematical framework is needed. Due to the complexity of the problems, however, there are different mathematical tools that fit to our needs and which highlights specifics aspects of a given field. In the recent research of the authors concerning global geometrical aspects of robust control theory it turns out that the proper mathematical language for the considerations seems to be a Jordan algebraic one, more precisely the framework of the Jordan pairs that is closely related to geometric, generalised projective geometric ideas. Moreover, this mathematical framework is closely related to the theory of unbounded and bounded symmetrical domains, where we can rediscover the concept of quadratic invariance, an efficient tool for the design of decentralized control systems. Since these mathematical tools seem to be completely unused in the previous control theoretical researches, in this paper we try to give a glimpse on the field. Our aim is to point out the main connections of this mathematical framework and control theory and to illustrate the potential of the framework for control problems.