Optimization problems for the controlled functional-differential equation with the mixed initial condition. The necessary conditions of optimality

In the work, for the functional-differential equation with two types controls and mixed initial condition, the optimization problem with general boundary conditions and functional is considered. The necessary conditions of optimality for initial and final moments, delay parameters, initial vector, initial and control functions are proved.The obtained results are specified for the optimal problem with fixed ends and integral functional, time optimal and linear problems. Besides, the optimal problem for the market relationship model is discussed, for which the necessary conditions of optimality are obtained.