Analytic Solution of a Delay Differential Equation Arising in Cost Functionals for Systems with Distributed Delays

TL;DR

This paper develops an analytic solution framework for a delay differential equation related to quadratic cost functionals in linear systems with delays, using an auxiliary ODE system and spectral analysis.

Contribution

It introduces a novel delay-free auxiliary ODE system with split-boundary conditions to solve delay differential equations in control systems.

Findings

Provides necessary and sufficient conditions for solution existence and uniqueness.

Establishes a tractable analytic solution method for delay differential equations.

Connects spectral properties of the system to solvability conditions.

Abstract

The solvability of a delay differential equation arising in the construction of quadratic cost functionals, i.e. Lyapunov functionals, for a linear time-delay system with a constant and a distributed delay is investigated. We present a delay-free auxiliary ordinary differential equation system with algebraically coupled split-boundary conditions, that characterizes the solutions of the delay differential equation and is used for solution synthesis. A spectral property of the time-delay system yields a necessary and sufficient condition for existence and uniqueness of solutions to the auxiliary system, equivalently the delay differential equation. The result is a tractable analytic solution framework to the delay differential equation.