  In this paper, we investigate the mean squared derivative cost functions that
arise in various applications such as in motor control, biometrics and optimal
transport theory. We provide qualitative properties, explicit analytical
formulas and computational algorithms for the cost functions. We also perform
numerical simulations to illustrate the analytical results. In addition, as a
by-product of our analysis, we obtain an explicit formula for the inverse of a
Wronskian matrix that is of independent interest in linear algebra and
differential equations theory.
