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<p>The focus of this work is the study of the controllability aspect of Gerasimov–Caputo fractional integro-differential equations with control inputs. The existence of the controlled solutions is proved by the use of the Schauder fixed-point theorem and the application of suitable assumptions. A numerical example is presented to give proof of the theoretical findings, along with a 3D graphical representation that shows the effect of time and control inputs on the system state. Also, a data-driven approach is proposed with the help of artificial neural networks to estimate the state trajectories more efficiently. The artificial neural network predictions are in a very close match with the numerical reference solutions and are able to capture the nonlinear dynamics and the memory-dependent behavior of the fractional system very well. This hybrid framework consisting of rigorous theoretical verification, numerical validation, graphical interpretation, and artificial neural network-based approximation provides a practical and computationally efficient methodology for analyzing and controlling complex fractional-order systems in engineering and applied sciences.</p>

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Controllability of Gerasimov–Caputo fractional integro-differential equations: a theoretical and data-driven approach

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Prabakaran Raghavendran,
  • Yamini Parthiban,
  • Haci Mehmet Baskonus,
  • Kamalendra Kumar

摘要

The focus of this work is the study of the controllability aspect of Gerasimov–Caputo fractional integro-differential equations with control inputs. The existence of the controlled solutions is proved by the use of the Schauder fixed-point theorem and the application of suitable assumptions. A numerical example is presented to give proof of the theoretical findings, along with a 3D graphical representation that shows the effect of time and control inputs on the system state. Also, a data-driven approach is proposed with the help of artificial neural networks to estimate the state trajectories more efficiently. The artificial neural network predictions are in a very close match with the numerical reference solutions and are able to capture the nonlinear dynamics and the memory-dependent behavior of the fractional system very well. This hybrid framework consisting of rigorous theoretical verification, numerical validation, graphical interpretation, and artificial neural network-based approximation provides a practical and computationally efficient methodology for analyzing and controlling complex fractional-order systems in engineering and applied sciences.