Optimality Conditions for Generalized Variable-Order Caputo Fractional Variational Problems

This study establishes a complete theoretical framework for a new class of fractional variational problems (FVPs) defined with Lagrangians dependent on left and right generalized variable-order Caputo fractional derivatives. The central result is the derivation of the fractional Euler-Lagrange equation, which provides a fundamental necessary condition for optimality. We significantly extend this result by generalizing the problem to include higher-order derivatives and multiple dependent variables. Furthermore, we develop necessary conditions for optimization subject to isoperimetric and holonomic constraints and characterize the associated natural boundary conditions. To complete the theoretical foundation, we also establish sufficient conditions for optimality under convexity assumptions. The practical relevance of our work is demonstrated through several analytical examples, which highlight the critical influence of the variable order and kernel function on the optimization process.