Akademik Veri Yönetim Sistemi
International Journal of Systems Science, 2026 (SCI-Expanded, Scopus)
This study introduces two- and three-dimensional optimal control problems characterised by the ψ-tempered fractional derivative and proposes a novel numerical methodology based on deep fractional-order Bernoulli optimisation to achieve their efficient solution. To this end, the problems under consideration are first reformulated as equivalent variational problems. Subsequently, a deep neural network employing the fractional-order Bernoulli functions and sinh as activation functions is utilised to approximate the state variable. To facilitate the effective implementation of the proposed method, we construct several integral operators of both integer and ψ-tempered fractional orders based on the basis functions derived from the deep neural network. These operators are discretised via the Fejér quadrature rule adapted to the ψ-tempered fractional calculus, ensuring stability and high-precision integration. The proposed approach converts the 2D/3D ψ-tempered fractional optimal control problem into an algebraic system using deep neural networks with integral operators and Gauss–Legendre integration, which is efficiently solved via Newton's method. The proposed method combines simple implementation, low computational cost, and high accuracy. Its effectiveness and robustness are validated through several representative 2D and 3D examples, confirming the method's applicability to complex fractional optimal control problems.