Neural network framework for controllability of fractional Volterra Fredholm integro-differential equations with state-dependent delay

Volume 40, Issue 3, pp 292--309 https://dx.doi.org/10.22436/jmcs.040.03.01

Publication Date: July 08, 2025 Submission Date: March 26, 2025 Revision Date: April 27, 2025 Accteptance Date: May 12, 2025

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Authors

P. Raghavendran - Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R\(\&\)D Institute of Science and Technology, Chennai-600062, Tamil Nadu, India. T. Gunasekar - Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R\(\&\)D Institute of Science and Technology, Chennai-600062, Tamil Nadu, India. - Department of Mathematics, Institute of Engineering and Technology, Srinivas University, Mukka, Mangaluru, Karnataka 574146, India. D. Baleanu - Department of Computer Science and Mathematics, Lebanese American University, Beirut 11022801, Lebanon. - Institute of Space Sciences-Subsidiary of INFLR, Magurele-Bucharest, Kalyani, West Bengal 741235, Romania. Sh. S. Santra - Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal 741235, India. D. Majumder - Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal 741235, India.

Abstract

This paper presents a new approach to analyzing the controllability of fractional Volterra-Fredholm integro-differential equations with state-dependent delay, characterized by the Caputo fractional derivative and governed by a semigroup of compact and analytic operators. Controllability results are derived using Schauder's fixed point theorem, addressing the challenges of fractional dynamics and state-dependent delays. A key innovation in this study is the integration of theoretical analysis with advanced computational methods, specifically Physics-Informed neural networks, to approximate solutions and verify controllability conditions. To validate the theoretical findings, a detailed example is provided, along with numerical simulations that confirm the convergence of solutions. Graphical representations offer additional insights into the solution dynamics, enhancing the understanding of the system's behavior. By combining mathematical rigor with machine learning techniques, this work establishes a computational framework for tackling complex fractional systems, paving the way for further exploration of their controllability.

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Keywords

• Controllability
• Schauder fixed point theorem
• physics-informed neural networks (PINNs)
• numerical simulations

MSC

•  34A08
•  45J05
•  65L20
•  93C23
•  68T07

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