Under-approximations of reachable sets and tubes have been receiving growing
research attention due to their important roles in control synthesis and
verification. Available under-approximation methods applicable to
continuous-time linear systems typically assume the ability to compute
transition matrices and their integrals exactly, which is not feasible in
general, and/or suffer from high computational costs. In this note, we attempt
to overcome these drawbacks for a class of linear time-invariant (LTI) systems,
where we propose a novel method to under-approximate finite-time forward
reachable sets and tubes, utilizing approximations of the matrix exponential
and its integral. In particular, we consider the class of continuous-time LTI
systems with an identity input matrix and uncertain initial and input values
belonging to full dimensional sets that are affine transformations of closed
unit balls. The proposed method yields computationally efficient
under-approximations of reachable sets and tubes, when implemented using
zonotopes, with first-order convergence guarantees in the sense of the
Hausdorff distance. To illustrate its performance, we implement our approach in
three numerical examples, where linear systems of dimensions ranging between 2
and 200 are considered