Conservative input-state-output systems with evolution on a multidimensional integer lattice

Joseph A. Ball, Cora Sadosky, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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" ℤ dd>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.

Original languageEnglish
Pages (from-to)133-198
Number of pages66
JournalMultidimensional Systems and Signal Processing
Issue number2
StatePublished - 1 Apr 2005


  • Conservative multidimensional discrete-time system
  • Energy balance relation
  • Initial value problem boundary condition at infinity
  • Shift-invariant sublattice

ASJC Scopus subject areas

  • Software
  • Signal Processing
  • Information Systems
  • Hardware and Architecture
  • Computer Science Applications
  • Artificial Intelligence
  • Applied Mathematics


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