- ▪Y. Rouchaleau, E.D. Sontag, "On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains", J. Comput. System Sci., vol. 18, no. 1, pp. 65–75, 1979. pdf
Abstract
This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay-differential systems.
- ▪E.D. Sontag, Y. Rouchaleau, "Sur les anneaux de Fatou forts", C. R. Acad. Sci. Paris Sér. A-B, vol. 284, no. 5, pp. A331–A333, 1977. pdf
Abstract
It is well known that principal rings are strong Fatou rings. We construct here a more general type of strong Fatou rings. We also prove that the monoid of divisor classes of a noetherian strong Fatou ring contains only the zero element, and that the dimension of such a ring is at most two.
- ▪E.D. Sontag, Y. Rouchaleau, "On discrete-time polynomial systems", Nonlinear Anal., vol. 1, no. 1, pp. 55–64, 1976. pdf
Abstract
Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.