A Brief Historical Note on Time-Delay Systems

Dynamic systems with delays arise naturally in multiple contexts, ranging from physiological and biological processes to engineering and economics. According to Bellman and Cooke (1963), systematic studies of this type of system began in the mid-20th century, when efforts were made to provide a solid mathematical foundation for delay differential equations, also known as functional differential equations. However, as early as Minorsky’s pioneering work  on ship course control (Minorsky, 1922), it was recognized that delays were responsible for unwanted oscillations and instabilities in the operation of control systems.

Over the following decades, the theory of time-delay systems was consolidated as an essential area of control theory. An important milestone was the introduction of Lyapunov–Krasovskii functionals (Krasovskii, 1956), which provided a general tool for the stability analysis of delay-type systems and which today constitute one of the cornerstones for the analysis and design of controllers for various classes of time-delay systems.

Bellman, R., & Cooke, K. L. (1963). Differential-difference equations. New York: Academic Press.

Minorsky, N. (1922). Directional stability of automatically steered bodies. Journal of the American Society for Naval Engineers, 34(2), 280–309. https://doi.org/10.1111/j.1559-3584.1922.tb04958.x

Krasovskii, N. N. (1956). Stability of motion: Applications of Lyapunov’s second method to differential systems and equations with delay. Stanford, CA: Stanford University Press.