Extremal solutions of differential inclusions via Baire category: A dual approach

Let F be a continuous multifunction on Rn 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]↦Rn with the following property: all solutions to the Cauchy problem x˙(t)∈Fw(t)(x(t)), x(0)=0, are also solutions to x˙(t)∈extF(x(t)). We prove that W is residual in C([0,T];Rn).