Appl Math Optim (2008) 58: 373–392
On Factorizations of Smooth Nonnegative
Matrix-Values Functions and on Smooth Functions
with Values in Polyhedra
Published online: 19 March 2008
© Springer Science+Business Media, LLC 2008
Abstract 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.
Keywords Finite-difference approximations · Polyhedra · Diagonally dominant
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, 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, 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.
The work was partially supported by NSF Grant DMS-0653121.
N.V. Krylov (
University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA