Conservative Input-State-Output Systems with Evolution on a Multidimensional Integer Lattice Academic Article uri icon


  • 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.

publication date

  • January 1, 2005