Taming Delays in Dynamical Systems: Unbounded Verification of Delay Differential Equations

Delayed coupling between state variables occurs regularly in technical dynamic systems, especially embedded control. As it consequently is omnipresent in safety-critical domains, there is an increasing interest in the safety verifications of systems modeled by Delay Differential Equations (DDEs). In this paper, we leverage qualitative guarantees for the existence of an exponentially decreasing estimation on the solutions to DDEs as established in classical stability theory, and present a quantitative method for constructing such delay-dependent estimations, thereby facilitating a reduction of the verification problem over an unbounded temporal horizon to a bounded one. Our technique builds on the linearization technique of non-linear dynamics and spectral analysis of the linearized counterparts. We show experimentally on a set of representative benchmarks from the literature that our technique indeed extends the scope of bounded verification techniques to unbounded verification tasks. Moreover our technique is easy to implement and can be combined with any automatic tool dedicated to bounded verification of DDEs.