Inner and Outer Approximating Flowpipes for Delay Differential Equations

Delay differential equations are fundamental for modeling networked control systems where the underlying network induces delay for retrieving values from sensors or delivering orders to actuators. They are notoriously difficult to integrate as these are actually functional equations, the initial state being a function. We propose a scheme to compute inner and outer-approximating flowpipes for such equations with uncertain initial states and parameters. Inner-approximating flowpipes are guaranteed to contain only reachable states, while outer-approximating flowpipes enclose all reachable states. We also introduce a notion of robust inner-approximation, which we believe opens promising perspectives for verification, beyond property falsification. The efficiency of our approach relies on the combination of Taylor models in time, with an abstraction or parameterization in space based on affine forms, or zonotopes. It also relies on an extension of the mean-value theorem, which allows us to deduce inner-approximating flowpipes, from flowpipes outerapproximating the solution of the DDE and its Jacobian with respect to constant but uncertain parameters and initial conditions. We present some experimental results obtained with our C++ implementation.