Nonlinear L-2-gain is a finite gain concept that generalises the notion of conventional (linear) finite L-2-gain to admit the application of L-2-gain analysis tools of a broader class of nonlinear systems. The computation of tight comparison function bounds for this nonlinear L-2-gain property is important in applications such as small gain design. This article presents an approximation framework for these comparison function bounds through the formulation and solution of an optimal control problem. Key to the solution of this problem is the lifting of an L-2-norm input constraint, which is facilitated via the introduction of an energy saturation operator. This admits the solution of the optimal control problem of interest via dynamic programming and associated numerical methods, leading to the computation of the proposed bounds. Two examples are presented to demonstrate this approach.