On the Evaluation of Cost Functionals, with Particular Emphasis on Time-delay Systems

It is useful to have closed-form expressions for the cost functional $∫0∞e2(t)dt$ where e ( t ) is a function representing error in a control system. The paper shows how the well-known results for delay-free systems may be extended to the time-delay case and a survey of various methods is given. Finally, an alternative derivation of the Routh—Hurwitz stability criteria for the delay-free case is given. This makes use of the close relationship between the above integrals and stability theory.