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The State Space Isomorphism Theorem for Discrete-Time Infinite-Dimensional Systems

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Abstract

It is well-known that the state space isomorphism theorem fails in infinite-dimensional Hilbert spaces: there exist minimal discrete-time systems (with Hilbert space state spaces) which have the same impulse response, but which are not isomorphic. We consider discrete-time systems on locally convex topological vector spaces which are Hausdorff and barrelled and show that in this setting the state space isomorphism theorem does hold. In contrast to earlier work on topological vector spaces, we consider a definition of minimality based on dilations and show how this necessitates certain definitions of controllability and observability.

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  1. Department of Mathematical Sciences, University of Bath, Bath, UK

    Amine N. Chakhchoukh & Mark R. Opmeer

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  1. Amine N. Chakhchoukh
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  2. Mark R. Opmeer
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Correspondence to Mark R. Opmeer.

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Chakhchoukh, A.N., Opmeer, M.R. The State Space Isomorphism Theorem for Discrete-Time Infinite-Dimensional Systems. Integr. Equ. Oper. Theory 84, 105–120 (2016). https://doi.org/10.1007/s00020-015-2251-4

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  • DOI: https://doi.org/10.1007/s00020-015-2251-4

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