In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.
ACM Transactions on Computer Systems
Computational Topology for Systems and Control
Scientific Computing

Collins, P.J. (2007). Optimal semicomputable approximations to reachable and invariant sets. ACM Transactions on Computer Systems, 41(1), 33–48.