Abstract
A fundamental object of study in both operator theory and system theory is a discrete-time conservative system (variously also referred to as a unitary system or unitary colligation). In this paper we introduce three equivalent multidimensional analogues of a unitary system where the “time axis” \({\mathbb Z}^{d}\), d>1, is multidimensional. These multidimensional formalisms are associated with the names of Roesser, Fornasini and Marchesini, and Kalyuzhniy–Verbovetzky. We indicate explicitly how these three formalisms generate the same behaviors. In addition, we show how the initial-value problem (including the possibility of “initial conditions at infinity”) can be solved for such systems with respect to an arbitrary shift-invariant sublattice as the analogue of the positive-time axis. Some of our results are new even for the d=1 case.
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First online version published in May 2005
*The authors were supported in part by a grant from the US-Israel Binational Science Foundation.
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Ball, J.A., Sadosky, C. & Vinnikov, V. Conservative Input–State–Output Systems with Evolution on a Multidimensional Integer Lattice. Multidim Syst Sign Process 16, 133–198 (2005). https://doi.org/10.1007/s11045-005-6861-x
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DOI: https://doi.org/10.1007/s11045-005-6861-x