Analysis of extremum value theorems for function spaces in optimal control under numerical uncertainty

The extremum value theorem for function spaces plays the central role in optimal control. It is known that computation of optimal control actions and policies is often prone to numerical errors which may be related to computability issues. The current work addresses a version of the extremum value theorem for function spaces under explicit consideration of numerical uncertainties. It is shown that certain function spaces are bounded in a suitable sense, i.e., they admit finite approximations up to an arbitrary precision. The proof of this fact is constructive in the sense that it explicitly builds the approximating functions. Consequently, existence of approximate extremal functions is shown. Applicability of the theorem is investigated for finite-horizon optimal control, dynamic programming and adaptive dynamic programming. Some possible computability issues of the extremum value theorem in optimal control are shown on counterexamples.