Computability of analog networks

We define a general concept of a network of analog modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u:T→A, and the input–output behaviour of the network with system parameters from A is modelled by a function  Φ:Ar×C[T,A]p→C[T,A]q(p,q>0,r≥0), where C[T,A] is the set of all continuous streams equipped with the compact-open topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle based on the fact that C[T,A] can be approximated locally by metric spaces. We show that if the module functions are continuous then so is the network function Φ. We analyse in detail two case studies involving mechanical systems. Finally, we introduce a custom-made concrete computation theory over C[T,A] and show that if the module functions are concretely computable then so is Φ.