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

• * Corresponding author: Stephen Campbell
• 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.

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

 Citation: • • 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 $\tau = 1.0.$

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

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

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 $\tau = 1.0.$

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 $\tau = 1.0.$

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

Figure 7.  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 

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

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

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

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

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

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

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

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

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

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

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