Under-Approximate Reachability Analysis for a Class of Linear Systems with Inputs

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