Let P be the class of optimal control problems having given fixed dimensions for states and controls, time interval [t0, t1], and control region, and assume each problem P in P is characterized by its initial state, control law, and integral objective functional: J=∫t0t1ƒ0[x(t), u(t), t] dt Perturbations P̃ of problems P in P are considered with respect to a certain topology τ on P and the L1 metric on the space of control functions, or controls. Essential uniqueness of an optimal control is introduced, and it is shown that any optimal control ũ∗, for a perturbation P̃ of a problem P having an essentially unique optimal control u∗, depends continuously upon P̃, that is, with respect to J and the L1 metric; i.e., given ε >; 0, there is a neighbourhood N of P in P such that PεN ⇒ d(u∗, ũ∗) < ε, where d represents the L1 metric. Also, it is shown that the optimal control for problems in P with a normal ‡‡See Section 8 for a definition. linear control law and a bounded convex polyhedron for the control region is essentially unique. We thus obtain a theorem that generalizes the type of result obtained by Kirillova in Izvestia VUZ, Matematika, No. 4 (5), 1958, pp. 113–126, which has been translated by L. W. Neustadt in SIAM Journal on Control, Vol. 1, No. 2, wherein Kirillova considered time-optimal instead of fixed-time problems but allowed only linear perturbations of a linear problem. The theorems here are also related to some more recent works, which are referenced.