In this work we study the optimal control problem for a class of nonlinear time-delaysystems via paratingent equation with delayed argument. We use an equivalence theorembetween solutions of differential inclusions with time-delay and solutions of paratingentequations with delayed argument. We study the problem of optimal control which minimizes acertain cost function. To show the existence of optimal control, we use the maintopological properties of the set solutions of paratingent equation with delayedargument

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a ${{C}^{1}}$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$-dimensional points is contained in an $n$-dimensional ${{C}^{1}}$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of $\text{G}$. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.