Polyhedral contractive sets for positive bidirectional associative memories

The paper proposes developments for the set-invariance scenarios applicable to continuous-time bidirectional associative memories (abbreviated as BAMs). The new results represent sufficient conditions for testing the existence of exponentially contractive invariant sets with polyhedral shape. Two theorems are stated, so as to cover the local, as well as the global cases of invariance. For both theorems, the test hypothesis is formulated in terms of a matrix equality with constraints; this equality incorporates concrete information about the BAM dynamics and the polyhedral set, respectively. From the point of view of the mathematical constructions, the BAM dynamics is investigated within a comparison setting that uses the trajectories of a positive linear system. The practical exploitation of the theoretical results is illustrated by a numerical example run in MATLAB, which also offers an intuitive support for the visualization of the key features characterizing the invariance property. The dynamics of discrete-time BAMs can be similarly addressed.