# State Space Consistency and Differentiability Conditions for a Class of Causal Dynamical Input-Output Systems - Mathematics > Dynamical Systems

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Abstract: A causal input-output system may be described by a function space for inputs,a function space for outputs, and a causal operator mapping the input spaceinto the output space. A particular representation of the state of such asystem at any instant has been defined as an operator from the space ofpossible future inputs to that of future outputs. This representation is calledthe natural state. The purpose of this report is to investigate additionalproperties of the natural state in two areas. The first area has to do with thepossibility of determining the input-output system from its natural state set.A counterexample where this is not possible is given. Sufficient conditions foridentifying the system from its natural state set are given. The results inthis area are mostly for time-invariant systems. There are also somepreliminary observations on reachability. The second area deals withdifferentiability properties involving the natural state inherited from theinput-output system, including differentiability of the natural state andnatural state trajectories. A differential equation representation is given.The results presented in this report may be considered as aids in modelingphysical systems because system identification from state set holds in manymodels and is only tacitly assumed; also, differentiability is a usefulproperty for many systems.

Author: ** Demetrios Serakos**

Source: https://arxiv.org/