Using the recently defined generalized derivative, we present a generalized formulation of variation of calculus, which includes the classical and conformable formulation as particular cases. In the first part of the article, through the properties of this generalized derivative, we discuss the generalized versions of the Bois–Reymond lemma, a Tonelli-type existence theorem, Euler–Lagrange equation, d’Alembert principle, du Bois–Reymond optimality condition and Noether’s theorem. In the second part, we discuss the Picard–Lindelöf theorem, Grönwall’s inequality, Pontryagin’s maximum principle and Noether’s principle for optimal control. We end with an application involving the time fractional Schrödinger equation.
"CALCULUS OF VARIATIONS AND OPTIMAL CONTROL WITH GENERALIZED DERIVATIVE." Rocky Mountain J. Math. 53 (5) 1337 - 1370, October 2023. https://doi.org/10.1216/rmj.2023.53.1337