{"ID":2829440,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.12111","arxiv_id":"2512.12111","title":"Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey","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.","short_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-Lio...","url_abs":"https://arxiv.org/abs/2512.12111","url_pdf":"https://arxiv.org/pdf/2512.12111v1","authors":"[\"Navid Mojahed\",\"Hooman Fatoorehchi\",\"Shima Nazari\"]","published":"2025-12-13T01:01:52Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\"]","methods":"[]","has_code":false}