On the geometry of the Pontryagin maximum principle in Banach spaces ∗ M. I. Krastanov † N. K. Ribarska ‡ Ts. Y. Tsachev § December 16, 2014 Abstract. A basic idea of the classical approach to obtain necessary optimality conditions in the calculus of variations and optimal control is to perturb with suitable variations the reference trajectory and then compare the unperturbed value of the cost functional with the perturbed one. Here we compare several classes of variations for the study of inﬁnite-dimensional optimal control problems. We prove an abstract result on a non-separation property of two closed sets and obtain as corollaries versions of the maximum principle under various assumptions. Illustrative examples indicating some limits of applicability of the existing variational techniques are presented. Key words. Pontryagin maximum principle, inﬁnite-dimensional prob- lems, control variations AMS subject classiﬁcations. 49K20, 49K27, 35F25 * This work has been partially supported by the Soﬁa University “St. Kliment Ohridski” under the contract No. 013/09.04.2014. † Faculty of Mathematics and Informatics, University of Soﬁa, James Bourchier Boul. 5, 1126 Soﬁa, Department of Biomathematics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., block 8, 1113 Soﬁa, Bulgaria (krast@math.bas.bg) ‡ Faculty of Mathematics and Informatics, University of Soﬁa, James Bourchier Boul. 5, 1126 Soﬁa, Department of Operations Research, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., block 8, 1113 Soﬁa, Bulgaria (ribarska@fmi.uni-soﬁa.bg) § Department of Operations Research, Institute of Mathematics and Informatics, Bul- garian Academy of Sciences, Acad. G. Bonchev str., block 8, 1113 Soﬁa, Bulgaria (tsachev@math.bas.bg) 1