Mesh refinement method for solving optimal control problems with nonsmooth solutions using jump function approximations

Abstract A warm start method is developed for efficiently solving complex chance constrained optimal control problems. The warm start method addresses the computational challenges of solving chance constrained optimal control problems using biased kernel density estimators and Legendre-Gauss-Radau collocation with an $hp$ adaptive mesh refinement method. To address the computational challenges, the warm start method improves both the starting point for the chance constrained optimal control problem, as well as the efficiency of cycling through mesh refinement iterations. The improvement is accomplished by tuning a parameter of the kernel density estimator, as well as implementing a kernel switch as part of the solution process. Additionally, the number of samples for the biased kernel density estimator is set to incrementally increase through a series of mesh refinement iterations. Thus, the warm start method is a combination of tuning a parameter, a kernel switch, and an incremental increase in sample size. This warm start method is successfully applied to solve two challenging chance constrained optimal control problems in a computationally efficient manner using biased kernel density estimators and Legendre-Gauss-Radau collocation.

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An Improved Adaptive hp Mesh Refinement Method in Solving Optimal Control Problems

10.21203/rs.3.rs-526882/v1 ◽

2021 ◽

Author(s):

Changxin Luo ◽

Jiong Li ◽

Chijun Zhou ◽

Humin Lei

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Optimal Control Problems ◽

Computing Time ◽

Mesh Refinement ◽

Pseudospectral Method ◽

Change Rate ◽

Refinement Method ◽

Effective Choice

Abstract The adaptive hp pseudospectral method is an effective choice in solving the optimal control problem. In order to improve the computing efficiency of the adaptive hp method, an improved adaptive hp mesh refinement method is proposed in this paper. There are two main cores in this method. The first is to refine the mesh in advance based on the change rate of the control, so that the algorithm can more efficiently deal with the optimal control problem with discontinuous control. And the second is to based on the accumulation of curvature values to ensure that those positions that need to be refined more can be quickly allocated to more mesh points. These two are combined with each other to improve the solving efficiency of the algorithm proposed in this paper, which shortens the required computing time. In addition, the convergence of the algorithm is proved in this paper. The simulation results show the effectiveness and superiority of the algorithm proposed in this paper.

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Mesh refinement strategy for optimal control problems

10.1063/1.4825560 ◽

2013 ◽

Cited By ~ 3

Author(s):

L. T. Paiva ◽

F. A. C. C. Fontes

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Mesh Refinement ◽

Control Problems ◽

Refinement Strategy

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A rapid-based improvement on some mesh refinement strategies in solving optimal control problems

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/dnz004 ◽

2019 ◽

Vol 37 (2) ◽

pp. 395-421

Author(s):

Maedeh Souzban ◽

Omid Solaymani Fard ◽

Akbar H Borzabadi

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Mesh Refinement ◽

Computational Cost ◽

The State ◽

Computational Time ◽

Control Problems ◽

Arc Length ◽

State Approximation ◽

Refinement Strategy

Abstract Recently, a mesh refinement strategy is presented on pseudospectral methods for solving optimal control problems by using the relative curvature of the state approximation to choose the type of discretization change in each iteration. Nevertheless, this criterion requires a large amount of computational cost in terms of CPU time. The main goal of this paper is to draw attention to select a suitable criterion with fewer computational cost. To this end, we use the arc length of the state approximation in the mesh interval based on the relative error estimate that was recently provided. We also update the number of mesh intervals and the location of mesh points according to the behaviour of the arc length. Indeed, by implementing this criterion, we do not need to solve an optimization problem anymore, and so significantly reduce the computational time as well as CPU times. Finally, we illustrate the accuracy, efficiency and ability of the arc length criterion in comparison with the curvature by offering some numerical examples.

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