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ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

Existence and Optimal Control of Hilfer Fractional Stochastic Pantograph Differential Equations

Journal of Vibration Testing and System Dynamics 10(1) (2026) 35--51 | DOI:10.5890/JVTSD.2026.03.003

Ayoub Louakar$^{1}$, Devaraj Vivek$^{2}$, Hamid Lmou$^{1}$, Khalid Hilal$^{1}$, Ahmed Kajouni$^{1}$

$^{1}$ Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco

$^{2}$ Department of Mathematics, PSG College of Arts and Science, Coimbatore-641014, India

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Abstract

In this article, the existence and optimal control of stochastic pantograph differential equation (SPDEs) involving the Hilfer fractional derivative (HFD) was investigated, in which a set of novel conditions were built to illustrate the existence via the Schaefer's fixed point theorem and uniqueness via the Banach fixed point theorem. The existence of optimal control pairs for the corresponding Lagrange control systems is then explored. Stochastic elements introduce randomness, capturing real-world unpredictability, while pantograph equations incorporate scaled past states. Moreover, an example was proposed to showcase the practical applicability of the theoretical results.

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