SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1520-1554, January 2021.
In this paper, we provide an essentially complete answer to the question of minimal initial data required to solve an overdetermined system of linear partial difference equations with real constant coefficients using the notion of characteristic sets. A characteristic set is a special subset of the domain with the defining property that for every solution trajectory of the system of equations, the knowledge of the solution trajectory restricted to this set uniquely determines the trajectory over the whole domain. We emphasize the fact that subsets which are sublattices and unions of finitely many parallel translates of such sublattices are best suited to answer the question of the minimality of initial data. We first provide an algebraic characterization of a sublattice to be a characteristic sublattice. The main result of this paper provides conditions under which a system admits a union of a sublattice and finitely many parallel translates of it as a characteristic set; an important condition is the rank of the sublattice being equal to the Krull dimension of the system. For the condition when the rank of the sublattice is strictly less than the Krull dimension of the system, we show that neither the sublattice nor a finite union of sublattices can be a characteristic set. For the case when the rank of the sublattice is strictly greater than the Krull dimension of the system, a union of the sublattice and finitely many parallel translates of it is a characteristic set. But, unlike the case when the rank of the sublattice is equal to the Krull dimension of the system, in this case a proper sublattice of the given sublattice exists which along with its finitely many parallel translates now qualify as a characteristic set. We also show that for a given overdetermined system of partial difference equations, a characteristic set of the form given by a union of a sublattice and finitely many parallel translates of it always exists.

中文翻译：

---

离散$ n $ -D系统中唯一表征每条轨迹所需的初始数据的最小化

SIAM控制与优化杂志，第59卷，第2期，第1520-1554页，2021年1月。
在本文中，我们提供了一个基本完整的答案，即使用特征集的概念来求解具有实常数系数的线性偏微分方程组的超定系统时所需的最少初始数据。特征集是具有定义属性的域的一个特殊子集，对于方程组的每个解轨迹，限定于此集合的解轨迹的知识将唯一地确定整个域的轨迹。我们强调这样一个事实，即子晶格和此类子晶格的有限多次平行平移的并集的子集最适合回答初始数据极少的问题。我们首先提供子格的代数表征，以将其作为特征子格。本文的主要结果提供了一个条件，在该条件下，系统允许子格的并集和有限次数的平行平移作为特征集。一个重要条件是子晶格的秩等于系统的Krull维。对于子格的秩严格小于系统的Krull维的条件，我们证明了子格或子格的有限并集都不能成为特征集。对于子晶格的秩严格大于系统的Krull维的情况，子晶格的并集及其有限的许多平行平移是一个特征集。但是，与子晶格的秩等于系统的Krull维的情况不同，在这种情况下，存在给定子格子的适当子格子，该子格子及其有限数量的并行平移现在可以视为特征集。我们还表明，对于给定的偏差分方程组的超定系统，始终存在由子格的并集和有限次的多次平行平移给出的形式的特征集。