On invariance and reachability on semialgebraic sets for linear dynamics

Reachability analysis is a powerful tool which is being used extensively and efficiently for the analysis and control of dynamical systems, especially when linear systems and convex sets are involved. In this note, we investigate whether exact or approximate reachability operations can be performed efficiently for the affine-semialgebraic setting, that is when we are dealing with general affine dynamics and basic semialgebraic sets. We show that it is partially true, we pinpoint the underlying challenges when this is not possible and indicate some directions for this case.