Chattering on optimal trajectories of Hölder problems

The problem of the minimization of the mean-square deviation from the origin is studied in the class of functions on a half-axis with derivatives satisfying the Hölder condition. The support of the solution is shown to be a finite interval and the optimal function makes countably many switches from intervals of maximum increase of the velocity to intervals of its maximum decrease. Switches accumulate to the right end-point of the support. An application of the results to the problem of precise constants in Kolmogorov-type inequalities for fractional derivatives is presented.