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1. Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

   https://doi.org/10.1115/1.4046831  

   Patnaik, Sansit; Semperlotti, Fabio (September 2020, Journal of Computational and Nonlinear Dynamics)

   Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems. 

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2. Applications of Distributed-Order Fractional Operators: A Review

   https://doi.org/10.3390/e23010110  

   Ding, Wei; Patnaik, Sansit; Sidhardh, Sai; Semperlotti, Fabio (January 2021, Entropy)

   null (Ed.)

   Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date. 

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3. A generalized fractional-order elastodynamic theory for non-local attenuating media

   https://doi.org/10.1098/rspa.2020.0200  

   Patnaik, Sansit; Semperlotti, Fabio (June 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell’s Law of refraction and of the corresponding Fresnel’s coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations. 

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4. Global sensitivity analysis of fractional-order viscoelasticity model

   Miles, P.R. (June 2019, SPIE Smart Structures + Nondestructive Evaluation)

   In this paper, we investigate hyperelastic and viscoelastic model parameters using Global Sensitivity Analysis(GSA). These models are used to characterize the physical response of many soft-elastomers, which are used in a wide variety of smart material applications. Recent research has shown the effectiveness of using fractional-order calculus operators in modeling the viscoelastic response. The GSA is performed using parameter subset selection (PSS), which quantifies the relative parameter contributions to the linear and nonlinear, fractional-order viscoelastic models. Calibration has been performed to quantify the model parameter uncertainty; however, this analysis has led to questions regarding parameter sensitivity and whether or not the parameters can be uniquely identified given the available data. By performing GSA we can determine which parameters are most influential in the model, and fix non-influential parameters at a nominal value. The model calibration can then be performed to quantify the uncertainty of the influential parameters. 

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5. End-to-End Learning Framework for Solving Non-Markovian Optimal Control

   Zhang, Xiaole; Zhang, Peiyu; Xiao, Xiongye; Li, Shixuan; Tzoumas, Vasileios; Gupta, Vijay; Bogdan, Paul (August 2025, International Conference on Machine Learning (ICML))

   Integer-order calculus fails to capture the long-range dependence (LRD) and memory effects found in many complex systems. Fractional calculus addresses these gaps through fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control tasks. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing a novel method for system identification and optimal control strategy in FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving theoretical bounds on the sample complexity for learning accurate control policies under fractional-order dynamics. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control. 

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