On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition (Q6620607)

In this paper, the authors investigate the following quasi-linear neutral functional-differential equation\N\[\Nx^\prime(t)=A(t, x(t))x^\prime(t-\sigma)+f(t, x(t), x(t-\tau_0), u_0(t)),\quad t\in [t_0, t_1],\N\]\Nwith the initial condition\N\[\Nx(t)=\phi_0(t),\quad t<t_0,\quad x(t_0)=x_{00}.\N\]\NThe authors establish an analytic relation between solutions of the considered Cauchy problem and a corresponding perturbed problem. This relation allows one to get an approximate solution of the perturbed equation and it plays an important role in proving the necessary conditions of optimality in the optimization problems.