ℓ2-stability: The cases of infinite dimensional discrete autonomous systems and 2-D autonomous systems

In this paper, we analyze ℓ2-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 ℓ2-stability of such systems. We then use the theory of ℓ2-stability, thus developed, to analyze ℓ2-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 ℓ2-stability of the above-mentioned type of systems.