Fully discrete schemes for monotone optimal control problems

TL;DR

This paper develops and analyzes fully discrete finite element schemes for monotone optimal control problems, proving convergence rates and demonstrating numerical implementation without requiring semiconcavity assumptions.

Contribution

It introduces a novel discretization approach for monotone control problems, establishing convergence orders and providing numerical examples without semiconcavity.

Findings

Convergence order of $(h + rac{k}{ oot h})^rac{ ext{H"older constant}}{ ext{value function}}$

Special parameter relations yield convergence of order $k^{2/3 rac{ ext{H"older constant}}{ ext{value function}}}$

Numerical implementation confirms theoretical results

Abstract

In this article we study a finite horizon optimal control problem with monotone controls. We consider the associated Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function. We consider the totally discretized problem by using the finite element method to approximate the state space $\Omega$. The obtained problem is equivalent to the resolution of a finite sequence of stopping-time problems. The convergence orders of these approximations are proved, which are in general $(h+\frac{k}{\sqrt{h}})^\gamma$ where $\gamma$ is the H\"older constant of the value function $u$. A special election of the relations between the parameters $h$ and $k$ allows to obtain a convergence of order $k^{\frac{2}{3}\gamma}$, which is valid without semiconcavity hypotheses over the problem's data. We show also some numerical implementations in an example.