On Factorizations of Smooth Nonnegative Matrix-values Functions and on Smooth Functions with Values in Polyhedra

We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron. 1. Motivation One of the main goals of the article is to understand what kind of optimal control problems of diffusion processes is covered by the results of [3] and [7], where the processes are given by Itô equations in a “special” form, such that in the corresponding Bellman equation the second order part is represented as the sum of second-order derivatives with respect to fixed vectors (independent of the control parameter) times squares of real-valued functions that are Lipschitz continuous with respect to the space variables. Roughly speaking the answer is that all control problems with twice continuously differentiable diffusion matrices fall into the scheme of [3] and [7] whenever property (A) holds: these matrices for all values of control and time and space variables belong to a fixed polyhedron in the set of symmetric nonnegative matrices. In the author’s opinion the control problems with property (A) are the only ones which admit finite-difference approximations with monotone schemes based on scaling of a fixed mesh. For functions w(z) given in a Euclidean space and vectors ξ in that space set w(ξ) = (ξ,∇w) = ∑ i ξwzi , w(ξ)(ξ) = ∑ i,j ξξwzizj . In many situations one needs to represent a d× d nonnegative symmetric matrix u as the square of a matrix or more generally as the product vv∗, where v is not necessarily a square matrix. If u = (uij) = vv∗ and v = (vik) and for each k we introduce the vector vk = (vik) ∈ Rd, then for any smooth 1991 Mathematics Subject Classification. 15A99, 65M06.