# On Hamilton–Jacobi–Bellman equations with convex gradient constraints

### Ryan Hynd

University of Pennsylvania, Philadelphia, USA### Henok Mawi

Howard University, Washington, USA

## Abstract

We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients.

## Cite this article

Ryan Hynd, Henok Mawi, On Hamilton–Jacobi–Bellman equations with convex gradient constraints. Interfaces Free Bound. 18 (2016), no. 3, pp. 291–315

DOI 10.4171/IFB/365