Over-approximating the forward reach sets of controlled dynamical systems subject to set-valued uncertainties is a common practice in systems-control engineering for the purpose of performance verification. However, specific algebraic and topological results for the geometry of such sets are rather uncommon even for simple linear systems such as the integrators. This work explores the geometry of the forward reach set of the integrator dynamics subject to compact setvalued uncertainties in its control inputs. Our contribution includes derivation of a closed-form formula for the support functions of these sets. This result, then enables us to deduce the parametric as well as the implicit equations describing the exact boundaries of these reach sets. Specifically, the implicit equations for the bounding hypersurfaces are shown to be given by vanishing of certain Hankel determinants. Finally, it is established that the integrator reach sets are semialgebraic as well as translated zonoids. These results should be useful to benchmark existing reach set over-approximation algorithms, and to help design new algorithms for the same.

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This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, assuming box-valued input uncertainties, we establish that this compact convex reach set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.