Abstract
Verification by simulation, based on covering the set of time-bounded trajectories of a dynamical system evolving from the initial state set by means of a finite sample of initial states plus a sensitivity argument, has recently attracted interest due to the availability of powerful simulators for rich classes of dynamical systems. System models addressed by such techniques involve ordinary differential equations (ODEs) and can readily be extended to delay differential equations (DDEs). In doing so, the lack of validated solvers for DDEs, however, enforces the use of numeric approximations such that the resulting verification procedures would have to resort to (rather strong) assumptions on numerical accuracy of the underlying simulators, which lack formal validation or proof. In this paper, we pursue a closer integration of the numeric solving and the sensitivity-related state bloating algorithms underlying verification by simulation, together yielding a safe enclosure algorithm for DDEs suitable for use in automated formal verification. The key ingredient is an on-the-fly computation of piecewise linear, local error bounds by nonlinear optimization, with the error bounds uniformly covering sensitivity information concerning initial states as well as integration error.
The first, third and fifth authors are supported partly by “973 Program” under grant No. 2014CB340701, by NSFC under grants 91418204 and 61502467, by CDZ project CAP (GZ 1023), and by the CAS/SAFEA International Partnership Program for Creative Research Teams. The second and fourth authors are supported partly by Deutsche Forschungsgemeinschaft within the Research Training Group “SCARE - System Correctness under Adverse Conditions” (DFG GRK 1765).
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Notes
- 1.
In general, the initial condition is represented by \(\mathbf {x}(t) =\xi _0(t)\), for \(t \in \left[ -r_k,0 \right] \), where \(\xi _0\in \mathcal {X} \subseteq C^0(\left[ -r_k,0 \right] , \mathbb {R}^n)\), \(C^0(\left[ -r_k,0 \right] , \mathbb {R}^n)\) stands for all continuous functions mapping from \(\left[ -r_k,0 \right] \) to \(\mathbb {R}^n\), \(\mathcal {X}\) is compact and bounded. So, we can let \(\varTheta = \cup _{\xi \in \mathcal {X}} \xi (\left[ -r_k,0 \right] )\). Clearly, \(\varTheta \) is compact and bounded.
- 2.
For ease of presentation, we demonstrate the approach on DDEs with one single delay, and it readily extends to that with multiple delays as discussed in Sect. 3.
- 3.
For a general initial condition \(\varvec{g}(t)\), \(\mathbf {y}\) is initialized as \(\mathbf {y}\leftarrow \llbracket \varvec{g}(-r),\varvec{g}(-r+\tau ),\ldots ,\varvec{g}(0) \rrbracket \).
- 4.
Available from http://lcs.ios.ac.cn/~chenms/tools/DDEChecker_v1.0.tar.bz2.
- 5.
Available from https://www.uni-oldenburg.de/en/hysat/.
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Chen, M., Fränzle, M., Li, Y., Mosaad, P.N., Zhan, N. (2016). Validated Simulation-Based Verification of Delayed Differential Dynamics. In: Fitzgerald, J., Heitmeyer, C., Gnesi, S., Philippou, A. (eds) FM 2016: Formal Methods. FM 2016. Lecture Notes in Computer Science(), vol 9995. Springer, Cham. https://doi.org/10.1007/978-3-319-48989-6_9
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