Two-dimensional descriptor systems

Abstract Linear descriptor systems are governed by dynamical equations subject to algebraic constraints. In the one-dimensional case, where the systems only depend on a single index, usually time, the Weierstrass canonical form splits up the state vector in two parts, a causal part, running forward in time, and a non-causal part, running backward. In this paper linear time-invariant autonomous descriptor systems in two-dimensions are discussed and the condition on the existence of a non-trivial solution is derived, together with an explicit formula for the output of such systems. It is shown that the output of the model can be related to a causal and a non-causal part in each of the dimensions of the model, running forward and backward in the various dimensions respectively. The results are obtained by requiring that the solutions, for states and outputs, which are defined on a two-dimensional grid, are path invariant and unique.