Discrete time schemes for optimal control problems with monotone controls

Philipp, Eduardo A.; Aragone, Laura S.; Parente, Lisandro A.

doi:10.1007/s40314-014-0149-4

Cited by 3 publications

(3 citation statements)

References 10 publications

(14 reference statements)

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“…The previously presented discretization allowed us to obtain important convergence results (see [13]). Nevertheless, in order to obtain numerical methods to estimate u h it is also necessary the discretization in the state variables.…”

Section: Fully Discrete Infinite Horizon Problemmentioning

confidence: 99%

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Fully discrete schemes for monotone optimal control problems

2016

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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 + k √ 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 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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Section: Fully Discrete Infinite Horizon Problemmentioning

confidence: 99%

“…We study a fully discrete scheme for the numerical resolution of the infinite horizon monotone optimal control problem, through the analysis of the associated finite horizon problem as in [13].…”

Section: Introductionmentioning

confidence: 99%

Fully discrete schemes for monotone optimal control problems

2016

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“…[13, Lemma 4.1] Under the hypotheses (1.3) and (1.4), we have|u(x, a) − u T (t, x, a)| ≤ M f λ e −λ(T −t) . (2.6)and the Theorem 2.1 [13,. Theorem 3.5] …”

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confidence: 99%

Fully discrete schemes for monotone optimal control problems

Philipp¹,

Aragone²,

Parente³

2014

Preprint

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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 + k √ 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 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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A Multilinear HJB-POD Method for the Optimal Control of PDEs on a Tree Structure

Kirsten,

Saluzzi

2024

J Sci Comput

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Optimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goal is to mitigate this crucial obstruction developing a version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests introducing novel pruning strategies that improve the efficacy of the method. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation.

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Discrete time schemes for optimal control problems with monotone controls

Cited by 3 publications

References 10 publications

Fully discrete schemes for monotone optimal control problems

Fully discrete schemes for monotone optimal control problems

Fully discrete schemes for monotone optimal control problems

A Multilinear HJB-POD Method for the Optimal Control of PDEs on a Tree Structure

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