Generalized dilations and numerically solving discrete-time homogeneous optimization problems

We introduce generalized dilations, a broader class of operators than that of dilations, and consider homogeneity with respect to this new class of dilations. For discrete-time systems that are asymptotically controllable and homogeneous (with degree zero) we propose a method to numerically approximate any homogeneous value function (solution to an infinite horizon optimization problem) to arbitrary accuracy. We also show that the method can be used to generate an offline computed stabilizing feedback law.