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 Ω)) and in (W^{1,p}(Ω)) for (p \in [1,\infty)) and (p = \infty). Finally, {numerical results} are presented to illustrate the approach on a test example.

💡 Research Summary

This paper introduces a novel methodology for approximating semiconcave functions while preserving their intrinsic structural properties. Semiconcave functions, which naturally arise as value functions in optimal control, differential games, and optimal transport, are characterized by the condition that the map x ↦ v(x) – (C/2)|x|² is convex for some constant C > 0. A key observation exploited by the authors is that any semiconcave function can be expressed as the pointwise infimum of a family of C² functions whose Hessians are uniformly bounded by C. This representation, while theoretically involving an uncountable index set, can be truncated to a finite collection without losing approximation accuracy.

The authors first formalize this truncation. By partitioning the domain Ω into a fine mesh of rectangles (or cells) and selecting a representative point y in each cell, they construct quadratic support functions ϕ_y(x) = v(y) + p_y·(x–y) + (C/2)|x–y|², where p_y belongs to the super‑differential D⁺v(y) and satisfies |p_y| ≤ L, the Lipschitz constant of v. The family {ϕ_y} is uniformly C², and the finite minimum v_n(x) = min_{i=1,…,n} ϕ_{y_i}(x) converges uniformly to v as the mesh is refined.

A major technical hurdle is that the minimum operator ψ_n(a) = min_{i≤n} a_i is not differentiable, which precludes the use of gradient‑based optimization or smooth analysis. To overcome this, the paper introduces a smooth approximation ψ_{n,ε} built from a regularized positive‑part function g_ε. The function g_ε ∈ C^{1,1}(ℝ) satisfies: (i) g_ε(x) ≥ 0, (ii) 0 ≤ g’_ε(x) ≤ 1, (iii) g’’ε(x) ≥ 0 a.e., (iv) ‖(·)⁺ – g_ε‖∞ ≤ ε, and (v) g’_ε converges pointwise to the indicator of