This paper is devoted to a novel approximation technique for optimal feedback control. We study a constrained Optimal Control Problem (OCP) in the context of a closed-loop dynamic system described by Ordinary Differential Equations (ODEs). The approximate optimal feedback control design we develop involves a specific <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> -relaxation technique (see [1]). We present here an initial research related to explicit approximations of the optimal control processes governed by the closed-loop dynamic systems. The obtained mathematical result has a potential to be extended to some concrete numerical solution algorithms for optimal feedback control problems.