Analytic Solution of a Delay Differential Equation Arising in Cost Functionals for Systems with Distributed Delays
This work offers a tractable analytic method for solving a specific delay differential equation, which is incremental for control theorists working on Lyapunov functionals for time-delay systems.
The paper provides an analytic solution framework for a delay differential equation arising in quadratic cost functionals for linear time-delay systems with constant and distributed delays, using an auxiliary ODE system with split-boundary conditions. A spectral condition ensures existence and uniqueness of solutions.
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.