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

Navid Mojahed, Hooman Fatoorehchi, Shima Nazari

TL;DR

This survey consolidates the theory and practice of fractional calculus in optimal control and differential games, unifying Caputo, Riemann–Liouville, and Grünwald–Letnikov derivatives and mapping memory effects to practical surrogates (diffusive/state augmentation and Oustaloup CRONE). It develops fractional versions of variational principles, PMP, and HJB/HJBI, and analyzes LQR, tracking, MPC, and FOPID designs under memory, highlighting numerical methods (time-domain, frequency-domain, and diffusive augmentations) and reproducibility-focused benchmarks. By detailing canonical problems, computational trade-offs, and open questions, the work provides a roadmap for deploying memory-rich control and game strategies in engineering applications. Key contributions include a structured taxonomy of FO representations, rigorous stability and controllability results, and a comprehensive comparison of numerical approaches with practical guidance on selection and implementation. The practical impact lies in enabling memory-aware controllers and game strategies with predictable performance, robust tuning, and scalable solvers across cyber-physical systems, energy devices, and biomedical applications.

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.

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

TL;DR

This survey consolidates the theory and practice of fractional calculus in optimal control and differential games, unifying Caputo, Riemann–Liouville, and Grünwald–Letnikov derivatives and mapping memory effects to practical surrogates (diffusive/state augmentation and Oustaloup CRONE). It develops fractional versions of variational principles, PMP, and HJB/HJBI, and analyzes LQR, tracking, MPC, and FOPID designs under memory, highlighting numerical methods (time-domain, frequency-domain, and diffusive augmentations) and reproducibility-focused benchmarks. By detailing canonical problems, computational trade-offs, and open questions, the work provides a roadmap for deploying memory-rich control and game strategies in engineering applications. Key contributions include a structured taxonomy of FO representations, rigorous stability and controllability results, and a comprehensive comparison of numerical approaches with practical guidance on selection and implementation. The practical impact lies in enabling memory-aware controllers and game strategies with predictable performance, robust tuning, and scalable solvers across cyber-physical systems, energy devices, and biomedical applications.

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.

Paper Structure

This paper contains 36 sections, 91 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Fractional Calculus
  3. Definitions and Properties
  4. Equivalent Representations
  5. Fractional Dynamical Systems
  6. State-space forms
  7. Existence, Uniqueness, and Continuous Dependence
  8. Stability Notions: Mittag--Leffler and Spectral Sectors
  9. Controllability and Observability (LTI, Commensurate)
  10. Optimality Principles for FO Systems
  11. Fractional Calculus of Variations
  12. Fractional Pontryagin Maximum Principle (PMP)
  13. Dynamic Programming and FO--HJB
  14. Fractional Differential Games (FDGs)
  15. Canonical Problems: LQR, Tracking, and Regulation
  16. ...and 21 more sections

Figures (1)

  • Figure 1: Graphical abstract of the survey. The figure summarizes the overall structure of the paper, linking modeling and taxonomy of fractional-order systems, optimality principles and numerical schemes, and their applications in control and game-theoretic settings, together with benchmark problems and directions for future research.