Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey

TL;DR

This survey comprehensively reviews fractional calculus methods in modeling, analysis, and control of systems with memory, covering theoretical foundations, numerical techniques, and practical applications in control and game theory.

Contribution

It unifies notation for various fractional derivatives, relates them to practical approximations, and discusses extensions to optimal control and game theory with memory effects.

Findings

Comparison of numerical schemes for fractional derivatives

Analysis of accuracy and complexity trade-offs in fractional control

Discussion of open problems in equilibria with memory

Abstract

Many physical, biological, and engineered systems exhibit memory effects that challenge Markovian models. Fractional calculus provides nonlocal operators to capture hereditary dynamics. This survey connects modeling, analysis, and controller/game design for systems with memory. We unify notation for Caputo, Riemann-Liouville, and Grunwald-Letnikov derivatives and relate them to practical approximations, including diffusive (sum-of-exponentials) state augmentation and frequency-domain realizations (e.g., Oustaloup). We review fractional extensions of the calculus of variations and the Pontryagin maximum principle, and dynamic-programming formulations with memory, including path-dependent HJB for optimal control and HJI for zero-sum games. We cover design tools such as LQR, MPC, and fractional-order PID, as well as fractional differential games with Nash, Stackelberg, and minimax equilibria. Computational approaches are compared across time-domain schemes, frequency-domain approximations, and diffusive augmentations, highlighting accuracy-complexity trade-offs and remedies for the curse of history (windowing and sum-of-exponentials). We conclude with applications and open problems on equilibria with memory, Isaacs-type conditions, constraint handling, and scalable solvers.