連続時間アファイン力学系の実現理論

抄録

In this paper, we establish the realization theorems of Continuous Time Affine Dynamical Systems which are defined as the subclass of (general) Dynamical Systems.
The Affine Dynamical Systems are dynamical systems whose state spaces are the Affine Representations of the concatenation Monoid Ω, called Affine Ω-module, and whose readout maps are affine maps.
Though the definition of Affine Dynamical System is given in the form of (general) dynamical systems and affine spaces, we convert it to a new form of Affine Dynamical Systems, called Normal Affine Dynamical Systems, which consist of linear space parts of affine spaces and the initial states 0. By this new form of Affine Dynamical Systems the realization theory of this class becomes almost the same as the realization theory of the Linear Representation Systems, except its initial object of the category of linear Ω-modules with affine input map. Then we obtain the existence and the uniqueness theorems of canonical Affine Dynamical Systems, i. e., any input response map can be realized by canonical Affine Dynamical Systems and they are isomorphic in the category of Affine Dynamical Systems.
Affine Dynamical Systems are quasi-reachable iff their state sets are the affine hulls of the strict reachable sets, and are canonical iff they are quasi-reachable and distinguishable. The definition of canonicality of Affine Dynamical Systems is different from that of Linear Representation Systems, hence we need to establish the realization theorems for Affine Dynamical Systems.
Note that the class of Affine Dynamical Systems contains Linear Systems and Inhomogenous Bilinear Systems as a subclass, and the realization theory of these classes can be discussed in the framework of Affine Dynamical Systems.