Fast Approximate Dynamic Programming for Input-Affine Dynamics

We propose two novel numerical schemes for the approximate implementation of the dynamic programming (DP) operation concerned with finite-horizon optimal control of discrete-time systems with input-affine dynamics. The proposed algorithms involve discretization of the state and input spaces and are based on an alternative path that solves the dual problem corresponding to the DP operation. We provide error bounds for the proposed algorithms, along with a detailed analysis of their computational complexity. In particular, for a specific class of problems with separable data in the state and input variables, the proposed approach can reduce the typical time complexity of the DP operation from
$\operatorname{\mathcal {O}}(XU)$
to
$\operatorname{\mathcal {O}}(X+U)$
, where
$X$
and
$U$
denote the size of the discrete state and input spaces, respectively. This reduction in complexity is achieved by an algorithmic transformation of the minimization in DP operation to an addition via discrete conjugation.