University of Tehran Press Journal of Computational Applied Mechanics 2423-6713 56 3 2025 07 01 Variational approach to optimal control constrained by fractal-fractional differential equations 602 610 102305 10.22059/jcamech.2025.396838.1504 EN Yue Cheng School of Information Engineering, Yango University, Fuzhou 350015, China Jia-Hong Zhu School of Information Engineering, Yango University, Fuzhou 350015, China Peng-Bin Luo School of Information Engineering, Yango University, Fuzhou 350015, China Yue Shen School of Science, Xi'an University of Architecture and Technology, Xi’an 710055, China JI-Huan He Soochow University School of Jia Yang, Zhejiang Shuren University, Shaoxing 312028, China Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamil Nadu, India 0000-0002-1636-0559 Journal Article 2025 06 06 The extant corpus of literature pertaining to optimal control problems with partial differential equation (PDE) constraints is extensive. This paper introduces a novel variational approach to optimal control problems constrained by fractal-fractional differential equations. Utilizing the shallow water wave as a case study, the semi-inverse method is employed to establish the variational formulation. This approach not only exemplifies a novel mode of thinking but also has significant ramifications for the field. This novel approach to optimal control paves a promising path for further research and provides researchers and practitioners with a novel perspective and potential avenues for further exploration. By exploring this alternative approach, researchers and practitioners can develop a more profound understanding of the fundamental nature of optimal control problems and identify more effective solutions for a wide range of applications. https://jcamech.ut.ac.ir/article_102305_05c7813f5e6a9a14fd4b1f088361cb18.pdf