T. Tadumadze^{∗}, N. Gorgodze^{∗∗}, and L. Alkhazishvili^{∗∗∗}

∗I. Vekua Institute of Applied Mathematics Tbilisi, Georgia ∗∗A. Tsereteli Kutaisi State University Kutaisi, Georgia ∗∗∗I. Javakhishvili Tbilisi State University Tbilisi, Georgia

VARIATION FORMULAS OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS AND THEIR APPLICATION TO OPTIMAL CONTROL PROBLEMS

For various classes of differential equations with deviating arguments with a continuous and discontinuous initial conditions the variation formulas of solutions are obtained. The continuous (discontinuous) initial condition means that values of an initial function and a trajectory always coinside (generally speaking, do not coinside) at an initial moment.

For optimal problems with non-fixed initial moment and with continuous and discontinuous initial conditions the necessary conditions of optimality are proved.