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arXiv · 1407.1790

Fully discrete schemes for monotone optimal control problems

Eduardo A. Philipp·Laura S. Aragone·Lisandro A. Parente
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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 $Ω$. 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 $(h+\frac{k}{\sqrt{h}})^γ$ where $γ$ is the Hölder 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 $k^{\frac{2}{3}γ}$, which is valid without semiconcavity hypotheses over the problem's data. We show also some numerical implementations in an example.

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Eduardo A. Philipp, Laura S. Aragone, Lisandro A. Parente. 2014-07-07. Fully discrete schemes for monotone optimal control problems. https://arxiv.org/abs/1407.1790

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