Structure Preserving Approximation of Semiconcave Functions

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of $C^2$ functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in $C(\bar \Omega)$ and in $W^{1,p}(\Omega)$ for $p \in [1,\infty)$ and $p = \infty$. Finally, {numerical results} are presented to illustrate the approach on a test example.