Linear dynamical systems over integral domains

The notions of constant, discrete-time, and linear dynamical systems over a commutative ring and their corresponding input/output maps are defined and studied. Classical stability theory is generalized to systems over fields complete with respect to a rank-one valuation. The resulting “p-adic stability theory” is used to solve the realization problem for matrix sequences over a broad class of integral domains, generalizing results first announced in Rouchaleau, Wyman, and Kalman [Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 3404–3406]