Perturbed nonlinear control problems with data depending on a vector parameter are considered. Using second-order sufficient optimality conditions, it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof exploits the close connections between solutions of a Riccati differential equation and shooting methods for solving the associated boundary value problem. Solution differentiability provides a firm theoretical basis for numerical feedback schemes that have been developed for computing neighbouring extremals. The results are illustrated by an example that admits two extremal solutions. Second-order sufficient conditions single out one optimal solution for which a sensitivity analysis is carried out.