In many control systems the hereditary effects have a considerable importance during the evolution of the process. These control systems need to be formulated by systems of delay-differential equations or delay-differential inclusions.;The mainstream in studying optimization problems consists of obtaining necessary optimality conditions for optimality. Necessary optimality conditions for optimal control systems governed by differential inclusions without delay are extensively discussed by a large number of publications. However, there are just a few papers devoted to systems of delay-differential inclusions, most of these papers concern delay-differential inclusions with the initial condition given by a single-valued mapping. To the best of our knowledge, We are not familiar with any necessary optimality conditions for control problems described by neutral functional-differential inclusions.;This work is devoted to the study of necessary optimality conditions of control systems governed by hereditary differential inclusions of delay type or hereditary differential inclusions of neutral type.;For hereditary differential inclusions of delay type we study the generalized Bolza problem with set-valued initial conditions and endpoint constraints. In contrast to previous publications on this topic, we deals with the problem involving a set-valued mapping in the initial condition which is specific for delay-differential systems. A choice of the initial function provides an additional source for optimizing the cost functional. For hereditary differential inclusions of neutral type we consider the Bolza problem governed by a constrained neutral functional-differential inclusion. Such control systems contain time-delays not only in state variables but also in velocity variables, which make them essentially more complicated than delay-differential or differential-difference inclusions.;To achieve our goal we employ the method of discrete approximations to the original problem. First we use the finite-difference to replace the derivative in the original system with appropriate approximations, this allows us to build a well-posed sequence of discrete optimization problems for time-delayed discrete inclusions with a strong convergence of optimal solutions, the obtained discrete optimization problems are intrinsically nonsmooth, but they fortunately can be handled by generalized differentiation tools. Then, use the extended Lagrange multiplier rule to derive necessary optimality conditions in delay-difference counterparts of the original problem. Finally, pass to the limit from discrete approximations to obtain necessary optimality conditions for the original problem.
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