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Delay differential equations (DDEs) play an important role in the modeling of dynamic processes. Delays arise in contemporary control schemes like networked distributed control and can cause deterioration of control performance, invalidating both stability and safety properties. This induces an interest in DDE especially in the area of modeling and verification of embedded control. In this article, we present an approach aiming at automatic safety verification of a simple class of DDEs against requirements expressed in a linear-time temporal logic. As requirements specification language, we exploit metric interval temporal logic (MITL) with a continuous-time semantics evaluating signals over metric spaces. We employ an over-approximation method based on interval Taylor series to enclose the solution of the DDE and thereby reduce the continuous-time verification problem for MITL formulae to a discrete-time problem over sequences of Taylor coefficients. We encode sufficient conditions for satisfaction as SMT formulae over polynomial arithmetic and use the iSAT3 SMT solver in its bounded model-checking mode for discharging the resulting proof obligations, thus proving satisfaction of time-bounded MITL specifications by the trajectories induced by a DDE. In contrast to our preliminary work in [44], we can verify arbitrary time-bounded MITL formulae, including nesting of modalities, rather than just invariance properties.

Model Checking Delay Differential Equations Against Metric Interval Temporal Logic