In this paper we consider the one—step and the two-step qussilinear optimal control problems with moving ends in the sense of the maximal speed and prove the existence theorems. We assume that admissible controls are functions belonging to $L_2 \[0, T\]$ and to the convex and compact set U, and that the initial set $M_0$ and the terminal set $M_T$ are compact subsets of $R^n$ in the one-step problem, and $M_0 \subset R^{n_1}$, $M_T \subset R^{n_2}$ in the two-step problem.