This paper deals with the optimal control problem for a class of nonlinear systems occurring in nonholonomic motion planning. This class extends a canonical form, originally considered by Brockett, which has generated great interest and a large amount of literature. The extension consists in allowing a linear drift in the dynamics. In contrast to most of the existing literature on this topic, where only necessary conditions are given, we provide sufficient conditions for global optimality for the above-mentioned problem. Our derivation employs the Lagrange-functional approach to optimal control (recently developed by Kosmol and Pavon), which leads to a relation between the optimality of a given control and the existence of a solution of a Riccati differential equation. As a-consequence of a more general analysis, we prove optimality of the given controls for the canonical systems studied by Brockett.