ℓ: The cases of infinite dimensional discrete autonomous systems and 2-D autonomous systems

Abstract In this paper, we analyze l 2 -stability of infinite dimensional discrete autonomous systems given in a state space form with state transition matrix being a Laurent polynomial matrix A ( σ , σ − 1 ) in the shift operator σ . We give sufficient conditions and necessary conditions for l 2 -stability of such systems. We then use the theory of l 2 -stability, thus developed, to analyze l 2 -stability of discrete 2-D autonomous systems. We achieve this by showing how a discrete 2-D autonomous system can be converted to an equivalent infinite dimensional state space discrete autonomous system, where the state transition matrix turns out to be a Laurent polynomial matrix in the shift operator. Finally, we provide some easy-to-check numerical tests for l 2 -stability of the above-mentioned type of systems.

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