For non-linear delay and quasi-linear neutral controlled functional differential equations with mixed initial condition and with distributed delay in controls, linear representations of the variation of solutions (formulas of variation) are obtained with respect to perturbations of the initial moment, of the initial vector, of the initial function, and of the control function. "Mixed initial conditions" means that at the initial moment some coordinates of the trajectory do not coincide with the corresponding coordinates of the initial function. On one hand, variation formulas are used in the proof of necessary optimality conditions [R.V. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, New York, London, 1978; N.M. Ogustoreli, Time-Delay Control Systems, Academic Press, New York, London, 1966; L.W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton Univ. Press, Princeton, 1976; G.L. Kharatishvili, T.A. Tadumadze, Variation formulas of solutions and optimal control problems for equations with retarded argument, J. Math. Sci. (NY) 104 (1) (2007) 1-175]; on the other hand, they allow one to obtain approximate solutions of perturbed equations in analytic form. Formulas of variation for non-linear delay differential equations without a control function are proved in [G.L. Kharatishvili, T.A. Tadumadze, Variation formulas for solution of a nonlinear differential equation with time delay and mixed initial condition, J. Math. Sci. (NY) 148 (3) (2008) 302-330]. Moreover, in the present paper, for delay and neutral optimal control problems with non-fixed initial moment and with mixed initial condition, the necessary conditions of optimality are obtained. One of them, the essential novelty, is a necessary condition of optimality for the initial moment containing the effect of the mixed initial condition. (C) 2008 Elsevier Ltd. All rights reserved.
