Examination of solving optimal control problems with delays using GPOPS-Ⅱ

John T. Betts; Stephen Campbell; Claire Digirolamo

doi:10.3934/naco.2020026

Article Contents

2021, Volume 11, Issue 2: 283-305. Doi: 10.3934/naco.2020026

This issue Previous Article Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation Next Article Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay

Examination of solving optimal control problems with delays using GPOPS-Ⅱ

1.
Applied Mathematical Analysis, 2478 SE Mirromont Pl., Issaquah, WA, 98027, USA
2.
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

^* Corresponding author: Stephen Campbell
^* Corresponding author: Stephen Campbell

Received: March 31, 2019

Revised: January 31, 2020

Early access: May 2020

Published: June 2021

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Abstract

There are a limited number of user-friendly, publicly available optimal control software packages that are designed to accommodate problems with delays. GPOPS-Ⅱ is a well developed MATLAB based optimal control code that was not originally designed to accommodate problems with delays. The use of GPOPS-Ⅱ on optimal control problems with delays is examined for the first time. The use of various formulations of delayed optimal control problems is also discussed. It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal control problem when used as a delayed boundary value solver of the necessary conditions.

Keywords:

Mathematics Subject Classification: Primary:49M37;Secondary:34K28, 49M25.

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Figure 1. Left: Solutions to (8) obtained by GPOPS-Ⅱw and dde23 and Right: solutions to (8) obtained by ode45w and dde23 for $ \sigma = -1.2 $ and $ \tau = 1.0. $

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Figure 2. State (left) and control (right) for (2) with $ \sigma = 1.2 $ using GPOPS-Ⅱm. Computed cost was 44.6641

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Figure 3. State (left) and control (right) for (2) with $ \sigma = 1.2 $ using GPOPS-Ⅱ on the MOS formulation. Computed cost was 43.4214. SOSD gave a similar appearing control and computed cost

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Figure 4. Left: Iterative states of GPOPS-Ⅱow for (9) and Right: states obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $ \sigma = -1.2 $ and $ \tau = 1.0. $

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Figure 5. Left: Iterative controls of GPOPS-Ⅱow for (9) and Right: controls obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $ \sigma = -1.2 $ and $ \tau = 1.0. $

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Figure 6. State (left) and control (right) obtained for (25) using GPOPS-Ⅱ and MOL

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Figure 7. State (left) and control (right) for (31) solving (26) using GPOPS-Ⅱm

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Figure 8. State (left) and control (right) from solving (26), Figure 8 is from [23]

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Figure 9. State (left) and control (right) for (28) using GPOPS-Ⅱm. Computed cost was 52.8417171

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Figure 10. State (left) and control (right) for (28) using SOSD. Computed cost was 53.27103

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Figure 11. State (left) and control (right) for (28) using the modified cost (29) with $ \alpha = 0.01 $ and also with SOSD on the original problem

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Figure 12. State (left) and control (right) for (30) using GPOPS-Ⅱm. Computed cost was 52.8417171

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Figure 13. State (left) and control (right) for (30) using SOSD. Computed cost was 56.187

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Figure 14. State (left) and control (right) for (1) using GPOPS-Ⅱm with prehistory a control variable. Computed cost was 52.8417171

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Figure 15. State (left) and control (right) for (9) with $ \sigma = -1.2, \tau = 1, $ found by solving the necessary conditions using GPOPS-Ⅱm

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Figure 16. State (left) and control (right) for (31) solving the necessary conditions (32) with GPOPS-Ⅱm

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Figure 17. State (left) and control (right) for (28) solving the necessary conditions with GPOPS-Ⅱm and also using SOSD on the original problem, $ \tau = 1, a = -1.14 $

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Figure 18. State (left) and control (right) for (30) solving the necessary conditions with GPOPS-Ⅱm

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