J Optim Theory Appl (2019) 180:303–320 https://doi.org/10.1007/s10957-018-1290-9 Optimal Control Problem for Bianchi Equation in Variable Exponent Sobolev Spaces Rovshan A. Bandaliyev 1,2 · Vagif S. Guliyev 1,2,3 · Ilgar G. Mamedov 4 · Yasin I. Rustamov 4 Received: 1 September 2017 / Accepted: 23 April 2018 / Published online: 3 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper, a necessary and sufficient condition, such as the Pontryagin’s maximum principle for an optimal control problem with distributed parameters, is given by the third-order Bianchi equation with coefficients from variable exponent Lebesgue spaces. The statement of an optimal control problem is studied by using a new version of the increment method that essentially uses the concept of the adjoint equation of the integral form. Keywords 3D optimal control · Pontryagin’s maximum principle · Bianchi equation · Goursat problem · Variable exponent Sobolev spaces Mathematics Subject Classification 37D30 · 49B20 · 49K20 B Rovshan A. Bandaliyev bandaliyevr@gmail.com Vagif S. Guliyev vagif@guliyev.com Ilgar G. Mamedov ilgar-mammadov@rambler.ru Yasin I. Rustamov Terlan56@mail.ru 1 Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan 2 S.M. Nikolskii Institute of Mathematics at RUDN University, Moscow, Russia 117198 3 Department of Mathematics, Ahi Evran University, Kirsehir, Turkey 4 Institute of Control Systems of NAS of Azerbaijan, Baku, Azerbaijan 123