- ▪E.D. Sontag, "The lattice of minimal realizations of response maps over rings", Math. Systems Theory, vol. 11, no. 2, pp. 169–175, 1977. pdf
Abstract
A lattice characterization is given for the class of minimal-rank realizations of a linear response map defined over a (commutative) Noetherian integral domain. As a corollary, it is proved that there are only finitely many nonisomorphic minimal-rank realizations of a response map over the integers, while for delay -differential systems these are classified by a lattice of subspaces of a finite-dimensional real vector space.
- ▪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, "On the internal realization of nonlinear behaviors", In Dynamical Systems, pp. 93–497, 1977.