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When dealing with two-dimensional (2-D) discrete state-space models, controllability properties are introduced in two different forms: a local form, which refers to single local states, and a global form, which instead pertains the infinite set of local states lying on a separation set. In this paper, these concepts are investigated in the context of 2-D positive systems by means of a graph theoretic approach. For all these properties, necessary and sufficient conditions, which refer to the structure of the digraph, are provided. While the global reachability index is bounded by the system dimension n, the local reachability index may far exceed the system dimension. Upper bounds on the local reachability index for some special classes of positive systems are finally derived. Index Terms--Controllability, finite memory systems, influence digraph, reachability, strong connectedness, two-dimensional (2-D) positive systems, zero controllability.

Bilinear systems in input-output form are introduced and represented by means of suitable rational functions in two indeterminates. Necessary and sufficient conditions for bounded-input/bounded-output (BIBO) stability are derived and compared with analogous results on two-dimensional systems stability. Several features of the output evolutions corresponding to finite support as well as to periodic input sequences are discussed and related to the polar structure of the rational functions describing the input-output maps. Index Terms - Bilinear systems, nonessential singularities, periodic dynamics, stability, two-dimensional systems.

Two-dimensional (2-D) codes are introduced as linear shift-invariant spaces of admissible signals on the discrete plane. Convolutional and, in particular, basic codes are characterized both in terms of their internal properties and by means of their input - output representations. The algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2-D decoders, free from catastrophic errors, as well as efficient syndrome decoders is considered. Some aspects of the state space implementation of 2-D encoders and decoders via (finite memory) 2-D system are discussed. Index Terms - Encoders and decoders of 2-D sequences, dual codes, behaviors, 2-D state models, inverse systems.