**Tamaz Tadumadze**

##
Variation Formulas of Solutions for Functional Differential Equations with
Several Constant Delays and Their Applications in Optimal Control Problems

**abstract:**

For nonlinear functional differential equations with several constant delays,
the theorems on the continuous dependence of solutions of the Cauchy problem on
perturbations of the initial data and on the right-hand side of the equation are
proved. Under the initial data we mean the collection of the initial moment,
constant delays, initial vector and initial function. Perturbations of the
initial data and of the right-hand side of the equation are small in a standard
norm and in an integral sense, respectively. Variation formulas of a solution
are derived for equations with a discontinuous initial and continuous initial
conditions. In the variation formulas, the effects of perturbations of the
initial moment and delays as well as the effects of continuous initial and
discontinuous initial conditions are revealed. For the optimal control problems
with delays, general boundary conditions and functional, the necessary
conditions of optimality are obtained in the form of equality or inequality for
the initial and final moments, for delays and an initial vector and also in the
form of the integral maximum principle for the initial function and control.

**Mathematics Subject Classification:**
34K99, 34K27, 49K21

**Key words and phrases:** Delay functional differential equations,
continuous dependence of solutions, variation formula of a solution, effect of
initial moment perturbation, effect of the discontinuous initial condition,
effect of the continuous initial condition, effect of constant delays
perturbations, optimal control problem with delays, necessary conditions of
optimality