On Arbitrary Assignability of Initial Conditions for a Discrete Autonomous $n$-D System

• 成果类型: Article
• 署名作者: Mukherjee, Mousumi; Pal, Debasattam
• 署名单位: Indian Institute of Technology System (IIT System); Indian Institute of Technology (IIT) - Bombay
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3206252
• 发表日期: 2023
• 页码: 4467-4473
• 关键词: n-D systems; algebraic analysis; characteristic sets; initial conditions; partial difference equations (PDEs)

摘要

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.