Estimating the growth of solutions of linear delayed difference and differential equations by alternating maximization

Numerical methods are proposed to quantify the magnitude of the growth reachable by solutions of systems of delayed linear difference and differential equations that are assumed to be asymptotically stable. A foundation based on an alternating maximization algorithm is established to address the discrete-time case. Following that, it is shown how to reuse this foundation for the continuous-time case, by converting to the discrete-time case through an approximation scheme that uses a backward differentiation formula (BDF) to produce a discretization in time. This indirect conversion approach raises new theoretical questions that are examined thoroughly. The proposed methods apply to systems with constant or variable coefficients. Numerical experiments are included to demonstrate their performance and reliability on several examples.