Extremal solutions to differential inclusions via Baire category: a dual approach

Let F be a continuous multifunction on IRn with compact convex values. For any vector w, let Fw(x) ⊆ F (x) be the subset of points which maximize the inner product with w. Call W the set of all continuous functions w: [0, T] 7 → IRn with the following property: all solutions to the Cauchy problem ẋ(t) ∈ Fw(t)(x(t)), x(0) = 0, are also solutions to ẋ(t) ∈ extF (x(t)). We prove that W is residual in C([0, T]; IRn).