Can Cybernetics and Fractional Calculus Be Partners?: Searching for New Ways to Solve Complex Problems

Abstract

Cybernetics addresses complex problems in space and time, but, surprisingly, it does not often employ fractional calculus. Here, we trace the history of cybernetics and fractional calculus to resolve this apparent paradox. Although the term cybernetics was first used in 1834 by the French physicist and mathematician Andre-Marie Ampere [2] to describe the science of government, its modern definition-control and communication in animals and machines-is traced to Norbert Wiener [3] (Figure 1). Since then, others have coined the words cyberspace and cyborg, and cybernetics is playing an increasing role in the study of human-machine systems and the analysis of the Internet of Things. Often defined as noninteger order integration and differentiation, the idea of fractional calculus first appeared in 1695 in a letter from Guillaume de L'Hopital to Gottfried Wilhelm Leibniz, but more recent interest can be traced to the work of Keith B. Oldham and Jerome Spanier [4] (Figure 2). Today, the techniques of fractional calculus are widely applied in many fields of science and engineering.