# 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
##### References:
 [1] Z. Bartoszewski and M. Kwapisz, On the convergence of waveform relaxation methods for differential-functional systems of equations, J. Math. Anal. Appl., 225 (1999), 478-496.  doi: 10.1006/jmaa.1999.6380.  Google Scholar [2] Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay-differential equations, SIAM J. Numerical Analysis, 38 (2011), 639-659.  doi: 10.1137/S003614299935591X.  Google Scholar [3] J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010. Google Scholar [4] J. T. Betts, N. Biehn, S. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods Ⅱ: computational experience, J. Comp. Appl. Math., 143 (2002), 237-261.  doi: 10.1016/S0377-0427(01)00509-X.  Google Scholar [5] J. T. Betts, S. L. Campbell and K. Thompson, Optimal control of delay partial differential equations, , in Control and Optimization with Differential-Algebraic Constraints, SIAM, (2012), 213–231. Google Scholar [6] J. T. Betts, S. L. Campbell and K. Thompson, Optimal control software for constrained nonlinear systems with delays, , Proc. IEEE Multi-Conference on Systems and Control (2011 MSC), Denver, (2011), 444–449. Google Scholar [7] J. T. Betts, S. L. Campbell and K. Thompson, Solving optimal control problems with control delays using direct transcription, Applied Numerical Mathematics, 108 (2016), 185-203.  doi: 10.1016/j.apnum.2015.12.008.  Google Scholar [8] N. Biehn, J. T. Betts, S. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods, J. Comp. Appl. Math., 125 (2000), 147-158.  doi: 10.1016/S0377-0427(00)00465-9.  Google Scholar [9] G. V. Bokov, Pontryagin's maximum principle of optimal control problems with time-delay, J. Mathematical Sciences, 172 (2011), 623–634. (Russian version: Fundam. Prikl. Mat., 15 (2009), Issue 5, 3–19.) doi: 10.1007/s10958-011-0208-y.  Google Scholar [10] S. L. Campbell, J. T. Betts and C. Digirolamo, Comments on initial guess sensitivity when solving optimal control problems using interior point methods, Numerical Algebra, Control, and Optimization, 10 (2020), 39-41.   Google Scholar [11] C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar [12] J. F. Frankena, Optimal control problems with delay, the maximum principle and necessary conditions, J. Engineering Mathematics, 9 (1975), 53-64.  doi: 10.1007/BF01535497.  Google Scholar [13] L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control subject to mixed state control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar [14] L. Göllmann and H. Maurer, Theory and application of optimal control problems with multiple delays, J. Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar [15] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.  Google Scholar [16] T. Koto, Method of lines approximation of delay differential equations, Computers & Mathematics with Applications, 48 (2004), 45-59.  doi: 10.1016/j.camwa.2004.01.003.  Google Scholar [17] C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar [18] C. Y. Liu, Z. Gong, H. W. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182.   Google Scholar [19] C. Liu, Z. Gong, K. L. Teo, R. Loxton and E. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar [20] C. Y. Liu, R. Loxton and and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems & Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001.  Google Scholar [21] C. Y. Liu, R. Loxton and K. L. Teo, Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306.  doi: 10.1007/s10589-013-9632-x.  Google Scholar [22] C. Y. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 63 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.  Google Scholar [23] M. Maleki and I Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Institute, 351 (2014), 811-839.  doi: 10.1016/j.jfranklin.2013.09.027.  Google Scholar [24] J. Mead and B. Zubik-Kowal, An iterated pseudospectral method for delay partial differential equations, Applied Numerical Mathematics, 55 (2005), 227-250.  doi: 10.1016/j.apnum.2005.02.010.  Google Scholar [25] M. A. Patterson and A. V. Rao, GPOPS Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), 1-37.  doi: 10.1145/2558904.  Google Scholar [26] H. Peng, X. Wang, S. Zhang and B. Chen, An iterative symplectic pseudospectral method to solve nonlinear state-delayed optimal control problems, Commun. Nonlinear. Sci. Numer. Simulat., 48 (2017), 95-114.  doi: 10.1016/j.cnsns.2016.12.016.  Google Scholar [27] A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: Gpops, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Transactions Mathematical Software, 37 (2010), 1-39.  doi: 10.1145/2558904.  Google Scholar [28] L. F. Shampine and S. Thompson, Solving DDEs in Matlab, Applied Numerical Mathematics, 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.  Google Scholar [29] A. Wäechter and L.T. Biegler, On the implementation of interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Prog., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [30] Z. Wang and L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete and Continuous Dynamical Systems, Series B, 13 (2010), 685-708.  doi: 10.3934/dcdsb.2010.13.685.  Google Scholar [31] D. Wu, Y. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.  Google Scholar [32] Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay equations in a given right half plane using a spectral method, J. Comp. and Appl. Math., 236 (2012), 2499-2514.  doi: 10.1016/j.cam.2011.12.009.  Google Scholar [33] Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, , Internal report TW 596, Department of Computer Science, K. U. Leuven, May, 2011. Available by download from http://twr.cs.kuleuven.be/research/software/delay-control/roots/+. doi: 10.1016/j.cam.2011.12.009.  Google Scholar

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##### References:
 [1] Z. Bartoszewski and M. Kwapisz, On the convergence of waveform relaxation methods for differential-functional systems of equations, J. Math. Anal. Appl., 225 (1999), 478-496.  doi: 10.1006/jmaa.1999.6380.  Google Scholar [2] Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay-differential equations, SIAM J. Numerical Analysis, 38 (2011), 639-659.  doi: 10.1137/S003614299935591X.  Google Scholar [3] J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010. Google Scholar [4] J. T. Betts, N. Biehn, S. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods Ⅱ: computational experience, J. Comp. Appl. Math., 143 (2002), 237-261.  doi: 10.1016/S0377-0427(01)00509-X.  Google Scholar [5] J. T. Betts, S. L. Campbell and K. Thompson, Optimal control of delay partial differential equations, , in Control and Optimization with Differential-Algebraic Constraints, SIAM, (2012), 213–231. Google Scholar [6] J. T. Betts, S. L. Campbell and K. Thompson, Optimal control software for constrained nonlinear systems with delays, , Proc. IEEE Multi-Conference on Systems and Control (2011 MSC), Denver, (2011), 444–449. Google Scholar [7] J. T. Betts, S. L. Campbell and K. Thompson, Solving optimal control problems with control delays using direct transcription, Applied Numerical Mathematics, 108 (2016), 185-203.  doi: 10.1016/j.apnum.2015.12.008.  Google Scholar [8] N. Biehn, J. T. Betts, S. L. Campbell and W. Huffman, Compensating for order variation in mesh refinement for direct transcription methods, J. Comp. Appl. Math., 125 (2000), 147-158.  doi: 10.1016/S0377-0427(00)00465-9.  Google Scholar [9] G. V. Bokov, Pontryagin's maximum principle of optimal control problems with time-delay, J. Mathematical Sciences, 172 (2011), 623–634. (Russian version: Fundam. Prikl. Mat., 15 (2009), Issue 5, 3–19.) doi: 10.1007/s10958-011-0208-y.  Google Scholar [10] S. L. Campbell, J. T. Betts and C. Digirolamo, Comments on initial guess sensitivity when solving optimal control problems using interior point methods, Numerical Algebra, Control, and Optimization, 10 (2020), 39-41.   Google Scholar [11] C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar [12] J. F. Frankena, Optimal control problems with delay, the maximum principle and necessary conditions, J. Engineering Mathematics, 9 (1975), 53-64.  doi: 10.1007/BF01535497.  Google Scholar [13] L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control subject to mixed state control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar [14] L. Göllmann and H. Maurer, Theory and application of optimal control problems with multiple delays, J. Industrial and Management Optimization, 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar [15] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198.  doi: 10.3934/jimo.2017042.  Google Scholar [16] T. Koto, Method of lines approximation of delay differential equations, Computers & Mathematics with Applications, 48 (2004), 45-59.  doi: 10.1016/j.camwa.2004.01.003.  Google Scholar [17] C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar [18] C. Y. Liu, Z. Gong, H. W. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182.   Google Scholar [19] C. Liu, Z. Gong, K. L. Teo, R. Loxton and E. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar [20] C. Y. Liu, R. Loxton and and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems & Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001.  Google Scholar [21] C. Y. Liu, R. Loxton and K. L. Teo, Optimal parameter selection for nonlinear multistage systems with time-delays, Computational Optimization and Applications, 59 (2014), 285-306.  doi: 10.1007/s10589-013-9632-x.  Google Scholar [22] C. Y. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 63 (2014), 957-988.  doi: 10.1007/s10957-014-0533-7.  Google Scholar [23] M. Maleki and I Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Institute, 351 (2014), 811-839.  doi: 10.1016/j.jfranklin.2013.09.027.  Google Scholar [24] J. Mead and B. Zubik-Kowal, An iterated pseudospectral method for delay partial differential equations, Applied Numerical Mathematics, 55 (2005), 227-250.  doi: 10.1016/j.apnum.2005.02.010.  Google Scholar [25] M. A. Patterson and A. V. Rao, GPOPS Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), 1-37.  doi: 10.1145/2558904.  Google Scholar [26] H. Peng, X. Wang, S. Zhang and B. Chen, An iterative symplectic pseudospectral method to solve nonlinear state-delayed optimal control problems, Commun. Nonlinear. Sci. Numer. Simulat., 48 (2017), 95-114.  doi: 10.1016/j.cnsns.2016.12.016.  Google Scholar [27] A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: Gpops, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Transactions Mathematical Software, 37 (2010), 1-39.  doi: 10.1145/2558904.  Google Scholar [28] L. F. Shampine and S. Thompson, Solving DDEs in Matlab, Applied Numerical Mathematics, 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.  Google Scholar [29] A. Wäechter and L.T. Biegler, On the implementation of interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Prog., 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [30] Z. Wang and L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete and Continuous Dynamical Systems, Series B, 13 (2010), 685-708.  doi: 10.3934/dcdsb.2010.13.685.  Google Scholar [31] D. Wu, Y. Bai and C. Yu, A new computational approach for optimal control problems with multiple time-delay, Automatica, 101 (2019), 388-395.  doi: 10.1016/j.automatica.2018.12.036.  Google Scholar [32] Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay equations in a given right half plane using a spectral method, J. Comp. and Appl. Math., 236 (2012), 2499-2514.  doi: 10.1016/j.cam.2011.12.009.  Google Scholar [33] Z. Wu and W. Michiels, Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, , Internal report TW 596, Department of Computer Science, K. U. Leuven, May, 2011. Available by download from http://twr.cs.kuleuven.be/research/software/delay-control/roots/+. doi: 10.1016/j.cam.2011.12.009.  Google Scholar
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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