  We consider minimum energy optimal control problem with time dependent
Lagrangian on the nonholonomic integrator and and find the analytical solution
using Sturm-Liouville theory. Furthermore, we also consider the minimum energy
problem on the Lie group $\mathbb{SO}(3)$ with time dependent Lagrangian.
  We show that the steering of nonholonomic integrator and generalized
nonholonomic integrator can be achieved by using various families of orthogonal
polynomials such as Chebyshev, Legendre and Jacobi polynomials apart from
trigonometric polynomials considered in the literature. Finally, we show how to
  find sub-optimal inputs using elements from a family of orthogonal functions
when the cost function is given by the $\mathcal{L}_1$ norm of the input.
