  To investigate solutions of (near-)optimal control problems, we extend and
exploit a notion of homogeneity recently proposed in the literature for
discrete-time systems. Assuming the plant dynamics is homogeneous, we first
derive a scaling property of its solutions along rays provided the sequence of
inputs is suitably modified. We then consider homogeneous cost functions and
reveal how the optimal value function scales along rays. This result can be
used to construct (near-)optimal inputs on the whole state space by only
solving the original problem on a given compact manifold of a smaller
dimension. Compared to the related works of the literature, we impose no
conditions on the homogeneity degrees. We demonstrate the strength of this new
result by presenting a new approximate scheme for value iteration, which is one
of the pillars of dynamic programming. The new algorithm provides guaranteed
lower and upper estimates of the true value function at any iteration and has
several appealing features in terms of reduced computation. A numerical case
study is provided to illustrate the proposed algorithm.
