Polynomial Chaos Expansions for the Stability Analysis of Uncertain DelayDifferential Equations

In the last few years the polynomial chaos theory of Wiener has been successfully applied to quantify uncertainty in many applications, since it may be a cheap alternative to Monte Carlo simulations. In this paper we introduce linear delay dierential equations with uncertain parameters, and we face both the well-posedness of the initial value problem and the stability by means of a suitable abstract reformulation. To quantify the eect of uncertainty on system stability, which is a crucial question in applications, we apply the polynomial chaos expansion to the stability indicator. The proposed numerical method combines the spectral discretization of the innitesimal generator and the stochastic collocation. Numerical results complete the paper.