A Hamilton-Jacobi Equation for Evaluating EEMI Propagation in a Computing System

Abstract

In this paper, we present a theoretical framework for modeling the empirically observed cascading of software failures on a complicated computing system exposed to extreme electromagnetic interference (EEMI). Our approach is to treat the temporal evolution of the electromagnetic coupling and the resultant cascading series of electromagnetic-induced faults as the "flow" in a dynamic fluid-mechanical system and thereby utilize aspects of the Navier Stokes and Hamilton-Jacobi equations to predict the rate of this flow. Therefore, inspired by the concepts of fluid dynamics [1], we include a diffusion term in the Hamilton-Jacobi-Isaacs (HJI) equation. We have considered two approaches. In one we apply a Taylor expansion to the optimality principle and consider additional terms; in the other scenario, we simply add a diffusion term in the form of a Laplacian applied to the cost function H(x,...) and some constant c, as it is present in the Navier-Stokes equation for incompressible flow. We provide numerical comparisons for both approaches with respect to the original HJI equation where the dynamical vector field corresponds to analytical models of a NOR logic gate. This model is a second-order differential equation that describes the behavior of the gate that incorporates a new term accounting for EEMI injection.