Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8586Pontryagin's Principle of Maximum for Linear Optimal Control Problems with Phase Constraints in Infinite Dimensional SpacesLonglaMКафедра дифференциальных уравнений и математической физики; Российский университет дружбы народов; Peoples' Friendship University of Russia-Peoples' Friendship University of Russia15042008451908092016Copyright © 2008,2008This paper presents the conditions of optimality for a problem with linear phase constraints in an infinite dimensional normal space with separated locally convex topology demonstrated using the works of M.F. Sukhinin in infinite dimensional normal spaces, his theory of differential equations in these spaces when functions are not Bochner-integrable and have no derivative of Gateaux. Problems with phase constraints were analyzed in finite spaces by many authors like L.S. Pontryagin, L. Graves, V.G. Boltyanskiy, R.V. Gamkrelidze, A.A. Milyutin, A.V. Dmitruk, N.P. Osmolovskij and others. Using the theory of differential equations of Prof. M.F. Sukhinin published in his monograph [1], applying the Gamkrelidze and Pontryagin's method illustrated in book [2], we enounced and proved theorems for linear mixed constraint in the separated locally convex space X.nonlinear optimizationtopologydifferential equationsconstraint problems