- The paper presents a unified framework integrating fractional calculus with optimal control and game theory, highlighting rigorous mathematical foundations and practical numerical methods.
- It details advanced numerical schemes such as convolution quadrature and CRONE approximations to address memory effects inherent in nonlocal dynamics.
- The survey provides benchmark templates and open research challenges, enabling reproducible experimentation for systems with nonlocal, memory-driven behaviors.
Fractional Calculus in Optimal Control and Game Theory: A Comprehensive Survey
Introduction and Motivation
The survey “Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey” (2512.12111) presents an authoritative synthesis at the intersection of fractional calculus (FC), optimal control (OC), and dynamic game theory. It systematically addresses modeling, analysis, numerical discretization, and control/game-theoretic design for systems exhibiting nonlocal memory, with an emphasis on rigorous characterization, reproducible experimentation, and open computational challenges.
Fractional dynamical systems, leveraging derivatives such as Caputo, Riemann--Liouville (RL), and Gr\"unwald--Letnikov (GL), provide parsimonious, physically meaningful models for phenomena with memory effects, including viscoelasticity, anomalous transport, and electrochemical dynamics. Standard Markovian models are often inadequate in such domains, and FC offers more compact and interpretable alternatives. This survey unifies notation, mathematical tools, and computational techniques for OC and differential games under fractional dynamics.
Unified Modeling and Mathematical Foundations
The survey rigorously formalizes key definitions: Caputo, RL, and GL derivatives, their equivalence, Laplace transforms, and their implications for initial/terminal data in control design. Caputo derivatives are preferred in OC due to their compatibility with standard initial conditions, while RL derivatives naturally accommodate hereditary models and non-integer initial data. The review highlights crucial distinctions: the lack of semigroup property for fractional derivatives, generalized integration by parts identities essential for optimality conditions, and the emergence of Mittag–Leffler functions as the natural analog of exponentials in time response and stability analysis.
The detailed taxonomy covers modeling with both commensurate and incommensurate fractional orders, augmented-state (diffusive/Prony) and frequency-domain (Oustaloup/CRONE) rational surrogates, and the translation of Caputo/RL dynamics to numerically tractable forms for both analysis and synthesis.
Fractional Optimality Principles
The FC extension of OC theory is comprehensively surveyed:
- Calculus of Variations & Optimality: Fractional Euler-Lagrange equations are detailed for both Caputo and RL action functionals, including boundary/transversality conditions with right-sided operators critical for variational analysis under memory.
- Pontryagin Maximum Principle (PMP): Time-nonlocal adjoint equations are formulated, with right-sided Caputo or RL derivatives encoding the history dependence in the costate dynamics. The necessary conditions are carefully distinguished, including scenarios with state/control constraints and terminal equalities.
- Dynamic Programming and Fractional HJB/HJBI: By moving beyond classical Markovian value functions, the survey rigorously details path-dependent HJB (and HJBI for zero-sum games) equations using fractional coinvariant derivatives. Both time-fractional and fully nonlocal path-dependent PDEs are reviewed, along with viscosity solution theory and state-of-the-art computational techniques for their solution.
Control and Game-Theoretic Design
Canonical problems—LQR, FOPID/CRONE, MPC, and fractional differential games—are exhaustively treated:
- FO-LQR and Tracking: The survey consolidates both ODE-augmented realizations (using diffusive or CRONE models) enabling classical Riccati-based synthesis and direct transcription methods yielding history-coupled QPs. Both approaches are shown to admit transparent accuracy-complexity trade-offs through the selection of mode count or time step.
- FOPID and Robust Control: The rationale for fractional PID controllers is linked to additional phase and gain degrees of freedom. CRONE loop shaping and frequency-domain design are contrasted with time-domain OC formulations; robust performance and iso-damped responses are explicitly addressed.
- Model Predictive Control (MPC): The memory overhead in FO-MPC due to “curse of history” is discussed with practical remedies: windowing, memory compression (SOE/fast algorithms), and performance/stability certification via Lyapunov and dissipativity arguments in the augmented setting.
- Differential Games: Fractional dynamic games are formulated with rigorous path-dependent value functionals. Existence, uniqueness, and Isaacs-type conditions are surveyed for both linear-quadratic and broader classes. Practical computational schemes (augmented-state TPBVPs, policy iteration, and value iteration on time-fractional HJBI) are discussed in detail.
Numerical Methods and Implementation
An exhaustive review of time- and frequency-domain numerics is provided. Key schemes include:
- Convolution Quadrature (CQ) and L1/L2-σ Schemes: Baseline stable and accurate discretizations are surveyed, including best practices for subdiffusion and error assessment.
- Diffusive/Prony Augmentation and Fast SOE: These methods permit memory-efficient, passivity-preserving state-space embedding of FO operators, crucial for practical real-time synthesis.
- CRONE/Oustaloup Rational Approximation: Band-limited rational models are shown to seamlessly interface with robust control and MPC.
Guidelines for initialization, mode/band selection, and maintaining physical properties (passivity, positive-realness) are explicit.
Applications Across Domains
The survey provides concrete templates for FO optimal control in electrochemical energy systems (battery ECMs, supercapacitors), viscoelasticity and structural damping, mechatronics/robotics, advanced thermal transport (non-Fourier models), and biomedical systems (fractional compartmental models). Each domain discussion links model structure to control/reconstruction pipeline, emphasizing practical parameter identification and reproducibility.
Benchmarks and Reporting
To promote apples-to-apples comparisons, the survey introduces a public benchmark suite for FO optimal control and game theory, prescribing canonical problem instances, recommended numerical realization settings, and explicit reporting checklists. This addresses a long-standing need in the field for reproducibility and quantitative evaluation of both accuracy and computational cost.
Open Problems and Research Challenges
The survey identifies fundamental open research directions, including:
- Existence and uniqueness under pointwise constraints and memory,
- Path-dependent dynamic programming, comparison theorems and viscosity solutions for general FO-HJB/HJBI,
- Intrinsic FO Riccati theory and characterization of optimal policies in native coordinates,
- Scalability of memory-aware MPC algorithms with certified error bounds,
- Robust and data-driven learning and identification respecting FO structure and stability sectors,
- Stochastic and uncertain FO dynamics, including risk-aware FO control and games.
Conclusion
This survey establishes foundational links among modeling, analysis, numerical methods, and design for FO OC and game-theoretic systems (2512.12111). Through meticulous theoretical synthesis and detailed computational workflows, it enables uniform treatment of memory-laden dynamics and delineates the actionable landscape for both rigorous research and practical applications. The curated benchmark suite and reporting guidelines are expected to accelerate progress and standardization in this rapidly evolving area, addressing both methodological rigor and applied control challenges in systems with inherent memory.
