Abstract
A discrete multidimensional system is the set of solutions to a system of linear partial difference equations defined on the lattice \(\mathbb {Z}^n\). This paper shows that it is determined by a unique coarsest sublattice, in the sense that the solutions of the system on this sublattice determine the solutions on \(\mathbb {Z}^n\); it is therefore the correct domain of definition of the discrete system. In turn, the defining sublattice is determined by a Galois group of symmetries that leave invariant the equations defining the system. These results find application in understanding properties of the system such as controllability and autonomy, and in its order reduction.
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Acknowledgements
We are very grateful to Ananth Shankar for his generous help with this paper. The first author acknowledges support from the ‘MATRICS’ Grant of the Science and Engineering Research Board, Govt. of India (Project File No. MTR/2019/000907). The second author is grateful to the Department of Electrical Engineering for its hospitality during many visits.
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Pal, D., Shankar, S. The coarsest lattice that determines a discrete multidimensional system. Math. Control Signals Syst. 34, 405–433 (2022). https://doi.org/10.1007/s00498-022-00318-1
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DOI: https://doi.org/10.1007/s00498-022-00318-1