Over- and Under-Approximating Reachable Sets for Perturbed Delay Differential Equations
This work provides a method for reachability analysis of perturbed DDEs, which is relevant for safety verification in systems with delays and uncertainties.
The paper extends reach set computation methods from ordinary differential equations to perturbed delay differential equations by exploiting the local homeomorphism property of solutions, enabling over- and under-approximations via boundary analysis. Three examples demonstrate the approach.
This note explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this note is a class of DDEs whose dynamics are subject to perturbations, and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over- and under-approximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.