FIFTH-RECURRENT GBK-FINSLER GEOMETRY AND ITS ENGINEERING RELEVANCE TO GEOMETRIC CONTROL AND ROBOTIC MOTION STABILITY

Saeedah M.  Baleedi; Adel Mohammed Ali  Al-Qashbari

doi:10.47372/ejua-ba.2025.4.480

Authors

Saeedah M. Baleedi Dept. of Mathematics, Education Faculty, University of Abyan, Zingibar, Yemen
Adel Mohammed Ali Al-Qashbari Dept. of Math's., Faculty of Educ. Aden, Univ. of Aden, Aden, Yemen

DOI:

https://doi.org/10.47372/ejua-ba.2025.4.480

Keywords:

Finsler geometry, Berwald covariant derivative, Kulkarni–Nomizu product, Robotic motion planning, Nonlinear dynamical systems

Abstract

This paper investigates the geometric structure of generalized BK-fifth recurrent Finsler spaces and analyzes the behavior of the associated curvature tensors under higher-order Berwald covariant derivatives. Using the Kulkarni–Nomizu product, several recurrence conditions are established for curvature expressions involving the K-, R-, H-, W-, and P-tensors. The results show that, under the condition λ_m=1/2, the fifth-order Berwald derivative of these tensors coincides with the fourth-order derivative, indicating a strong form of high-order geometric invariance. From an engineering perspective, such invariance provides a rigorous mathematical foundation for Finsler-based modeling in geometric control and robotic motion planning, where stable curvature structures enhance robustness in nonlinear stabilization, trajectory tracking, and anisotropic path optimization.