# American Institute of Mathematical Sciences

June  2021, 11(2): 283-305. doi: 10.3934/naco.2020026

## 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

Received  April 2019 Revised  February 2020 Published  June 2021 Early access  May 2020

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.

Citation: John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
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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 $\tau = 1.0.$
State (left) and control (right) for (2) with $\sigma = 1.2$ using GPOPS-Ⅱm. Computed cost was 44.6641
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
Left: Iterative states of GPOPS-Ⅱow for (9) and Right: states obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $\sigma = -1.2$ and $\tau = 1.0.$
Left: Iterative controls of GPOPS-Ⅱow for (9) and Right: controls obtained by SOSD, GPOPS-Ⅱm, and control parameterization with $\sigma = -1.2$ and $\tau = 1.0.$
State (left) and control (right) obtained for (25) using GPOPS-Ⅱ and MOL
State (left) and control (right) for (31) solving (26) using GPOPS-Ⅱm
]">Figure 8.  State (left) and control (right) from solving (26), Figure 8 is from [23]
State (left) and control (right) for (28) using GPOPS-Ⅱm. Computed cost was 52.8417171
State (left) and control (right) for (28) using SOSD. Computed cost was 53.27103
State (left) and control (right) for (28) using the modified cost (29) with $\alpha = 0.01$ and also with SOSD on the original problem
State (left) and control (right) for (30) using GPOPS-Ⅱm. Computed cost was 52.8417171
State (left) and control (right) for (30) using SOSD. Computed cost was 56.187
State (left) and control (right) for (1) using GPOPS-Ⅱm with prehistory a control variable. Computed cost was 52.8417171
State (left) and control (right) for (9) with $\sigma = -1.2, \tau = 1,$ found by solving the necessary conditions using GPOPS-Ⅱm
State (left) and control (right) for (31) solving the necessary conditions (32) with GPOPS-Ⅱm
State (left) and control (right) for (28) solving the necessary conditions with GPOPS-Ⅱm and also using SOSD on the original problem, $\tau = 1, a = -1.14$
State (left) and control (right) for (30) solving the necessary conditions with GPOPS-Ⅱm
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