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On Arbitrary Assignability of Initial Conditions for a Discrete Autonomous n-D System
Journal
IEEE Transactions on Automatic Control
ISSN
00189286
Date Issued
2023-07-01
Author(s)
Mukherjee, Mousumi
Pal, Debasattam
Abstract
The issue of initial/boundary conditions for a general system of partial difference equations (PDEs) - called a discrete n-D system - is resolved through the notion of characteristic sets: the restriction of a solution trajectory to a minimal characteristic set can be considered to be the initial/boundary condition to the corresponding trajectory. In the recent paper (Mukherjee and Pal, 2021), it has been shown that every autonomous n-D system admits a minimal characteristic set that is the union of a sublattice (of rank equal to the Krull dimension of the system) and its finitely many parallel translates. Treating the restrictions of every solution trajectories to such a characteristic set as initial/boundary conditions, in this article, we provide a full parametrization of the set of all initial/boundary conditions of a discrete autonomous n-D system. The key to this parametrization is that the set of allowable initial/boundary conditions itself can be viewed as a discrete d-D system, where d is the Krull dimension of the system: we prove this result here. An upshot of this parametrization is the answer to free assignability of these initial/boundary conditions.
Subjects