J Optim Theory Appl DOI 10.1007/s10957-016-0873-6 Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives Teodor M. Atanackovi´ c 1 · Marko Janev 2 · Stevan Pilipovi´ c 3 · Dušan Zorica 2,4 Received: 30 December 2014 / Accepted: 11 January 2016 © Springer Science+Business Media New York 2016 Abstract Two variational problems of finding the Euler–Lagrange equations corre- sponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional deriv- ative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler–Lagrange equations is obtained. It is shown that the sequence of approximated Euler–Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one. Keywords Complex-order fractional variational problems · Euler–Lagrange equations · Expansion formula · Weak convergence B Dušan Zorica dusan_zorica@mi.sanu.ac.rs Teodor M. Atanackovi´ c atanackovic@uns.ac.rs Marko Janev markojan@mi.sanu.ac.rs Stevan Pilipovi´ c pilipovic@dmi.uns.ac.rs 1 Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia 2 Mathematical Institute, Serbian Academy of Arts and Sciences, Beograd, Serbia 3 Department of Mathematics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia 4 Department of Physics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia 123