Loss of the Sturm–Liouville Property of Time-Varying Second-Order Differential Equations in the Presence of Delayed Dynamics

Abstract

This paper considers a nominal undelayed and time-varying second-order Sturm–Liouville differential equation on a finite time interval which is a nominal version of another perturbed differential equation subject to a delay in its dynamics. The nominal delay-free differential equation is a Sturm–Liouville system in the sense that it is subject to prescribed two-point boundary conditions. However, the perturbed differential system is not a Sturm–Liouville system, in general, due to the presence of delayed dynamics. The main objective of the paper is to investigate the loss of the boundary values of the Sturm–Liouville nominal undelayed system in the presence of the delayed dynamics. Such a delayed dynamics is considered a perturbation of the nominal dynamics related to the Sturm–Liouville system with given two-point boundary values. In particular, this loss of the Sturm–Liouville exact tracking of the prescribed two-point boundary values might happen because the nominal boundary values may become lost by the state trajectory solution in the presence of delays, related to the undelayed case, due to the presence of the delayed dynamics. The worst-case error description of the deviation of the two-point boundary values of the current perturbed differential with respect to those of the nominal Sturm–Liouville system is characterized in terms of error norms related to the nominal system. Under sufficiently small deviations of the parameterization of the perturbed system with respect to the nominal one, such a worst-error characterization makes the current perturbed system an approximate Sturm–Liouville system of the nominal undelayed one.