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The solvability of a delay differential equation arising in the constructionof quadratic cost functionals, i.e. Lyapunov functionals, for a lineartime-delay system with a constant and a distributed delay is investigated. Wepresent a delay-free auxiliary ordinary differential equation system withalgebraically coupled split-boundary conditions, that characterizes thesolutions of the delay differential equation and is used for solutionsynthesis. A spectral property of the time-delay system yields a necessary andsufficient condition for existence and uniqueness of solutions to the auxiliarysystem, equivalently the delay differential equation. The result is a tractableanalytic solution framework to the delay differential equation.

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