doi: 10.3934/dcdss.2020108

A time-scaling technique for time-delay switched systems

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

2. 

Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China

* Corresponding author: Changjun Yu

Received  May 2018 Revised  August 2018 Published  September 2019

Fund Project: This work is supported by National Natural Science Foundation of China (NSFC grant number 11871039, 11771275)

In this paper, we consider a class of optimal control problems governed by nonlinear time-delay switched systems, in which the system parameters and switching times between different subsystems are decision variables to be optimized. We propose a new computational approach to deal with the computational difficulties caused by variable switching times. The original time-delay switched system is firstly transformed into an equivalent switched system defined on a new time horizon where the switching times are fixed, but each of the subsystems contain a variable time-delay that depends on the durations of each sub-system in the original system. By deriving the analytical form for the variable time-delay in the new time horizon, we can solve the new time-delay switched system. Then, gradient-based optimization algorithm can be applied to solve the equivalent problem efficiently. Numerical results show that this new approach is effective.

Citation: Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020108
References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262. Google Scholar

[2]

N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. Google Scholar

[3]

I. AreaF. NdairouJ. J. NietoC. J. Silva and D. F. M. Torres, Ebola model and optimal control with vaccination constraints, Journal of Industrial and Management Optimization, 14 (2017), 427-446. doi: 10.3934/jimo.2017054. Google Scholar

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L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013. Google Scholar

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F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999.Google Scholar

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Q. ChaiR. LoxtonK. L. Teo and and C. Yang, A max-min control problem arising in gradient elution chromatography, Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144. doi: 10.1021/ie202475p. Google Scholar

[7]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471. Google Scholar

[8]

Q. ChaiR. LoxtonK. L. Teo and C. H. Yang, Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 17 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015. Google Scholar

[9]

Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, Journal of Industrial and Management Optimization, 14 (2018), 913-930. doi: 10.3934/jimo.2017082. Google Scholar

[10]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No.1. Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7. Google Scholar

[11]

G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numerical Algebra, Control and Optimization, 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619. Google Scholar

[12]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413. Google Scholar

[13]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843. Google Scholar

[14]

Z. H. GongC. 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

[15]

K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855. Google Scholar

[16]

C. Y. Kaya and J. L. Noakes, Computational method for time-optimal Switching Control, Journal of Optimization Theory and Applications, 117 (2003), 69-92. doi: 10.1023/A:1023600422807. Google Scholar

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H. W. J. Lee and K. H. Wong, Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints, Optimization Methods and Software, 21 (2006), 679-691. doi: 10.1080/10556780500142306. Google Scholar

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Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar

[20]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 18 (2011), 59-76. doi: 20.500.11937/48414. Google Scholar

[21]

C. Y. LiuR. 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 and Z. H. Gong, Modelling and optimal control of a time-delayed switched system in fed-batch process, Journal of the Franklin Institute, 351 (2014), 840-856. doi: 10.1016/j.jfranklin.2013.09.014. Google Scholar

[23]

C. Y. LiuZ. H. GongK. L. TeoJ. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar

[24]

C. Y. LiuZ. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368. doi: 10.1016/j.apm.2017.09.007. Google Scholar

[25]

C. Y. LiuR. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7. Google Scholar

[26]

R. LoxtonQ. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pacific Journal of Optimization, 10 (2014), 537-560. doi: 20.500.11937/19892. Google Scholar

[27]

R. LoxtonK. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2018), 2923-2929. doi: 10.1016/j.automatica.2008.04.011. Google Scholar

[28]

R. LoxtonK. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar

[29]

R. LoxtonK. L. TeoV. Rehbock and K. F. C. Yiu, Control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029. Google Scholar

[30]

R. LoxtonK. L. TeoV. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031. Google Scholar

[31]

R. Luus, Use of Luus-Jaakola optimization procedure for singular optimal control problems, Nonlinear Analysis, Theory, Methods & Applications, 47 (2001), 5647-5658. doi: 10.1016/S0362-546X(01)00666-6. Google Scholar

[32]

H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM Journal on Control Optimization, 42 (2004), 2239-2263. doi: 10.1137/S0363012902402578. Google Scholar

[33]

J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. Google Scholar

[34]

M. SchlegelK. StockmannT. Binder and W. Marquardt, Dynamic optimization using adaptive control vector parameterization, Computers and Chemical Engineering, 29 (2005), 1731-1751. doi: 10.1016/j.compchemeng.2005.02.036. Google Scholar

[35]

A. Siburian and V. Rehbock, Numerical procedure for solving a class of singular optimal control problems, Optimization Methods and Software, 19 (2004), 413-426. doi: 10.1080/10556780310001656637. Google Scholar

[36]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. Google Scholar

[37]

L. Y. WangW. H. GuiK. L. TeoR. Loxton and C. H. Yang, Optimal control problems arising in the zinc sulphate electrolyte purification process, Journal of Global Optimization, 54 (2012), 307-323. doi: 10.1007/s10898-012-9863-x. Google Scholar

[38]

L. Y. WangW. H. GuiK. L. TeoR. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705. Google Scholar

[39]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972. Google Scholar

[40]

K. H. WongL. S. Jennings and F. Benyah, The control parametrization enhancing transform for constrained time–delayed optimal control problems, ANZIAM Journal, 43 (2002), 154-185. doi: 10.21914/anziamj.v43i0.469. Google Scholar

[41]

S. F. WoonV. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594. doi: 10.1002/oca.1015. Google Scholar

[42]

C. Z. Wu and K. L. Teo, Optimal impulsive control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. Google Scholar

[43]

C. Z. WuK. L. TeoR. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. doi: 10.1016/j.aml.2005.11.018. Google Scholar

[44]

X. Xiang and Y. Peng, Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Management Optimization, 4 (2008), 17-32. doi: 10.3934/jimo.2008.4.17. Google Scholar

[45]

C. J. YuB. LiR. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7. Google Scholar

[46]

C. J. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901. doi: 10.1007/s10957-015-0783-z. Google Scholar

[47]

C. J. YuK. L. TeoL. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar

[48]

Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693. doi: 10.1021/ie200996f. Google Scholar

show all references

References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262. Google Scholar

[2]

N. U. Ahmed, Elements of Finite-Dimensional Systems and Control Theory, Longman Scientific and Technical, New York, 1988. Google Scholar

[3]

I. AreaF. NdairouJ. J. NietoC. J. Silva and D. F. M. Torres, Ebola model and optimal control with vaccination constraints, Journal of Industrial and Management Optimization, 14 (2017), 427-446. doi: 10.3934/jimo.2017054. Google Scholar

[4]

L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013. Google Scholar

[5]

F. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization, Ph.D thesis, Universita di Firenze, 1999.Google Scholar

[6]

Q. ChaiR. LoxtonK. L. Teo and and C. Yang, A max-min control problem arising in gradient elution chromatography, Industrial and Engineering Chemistry Research, 51 (2012), 6137-6144. doi: 10.1021/ie202475p. Google Scholar

[7]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471. Google Scholar

[8]

Q. ChaiR. LoxtonK. L. Teo and C. H. Yang, Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 17 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015. Google Scholar

[9]

Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems, Journal of Industrial and Management Optimization, 14 (2018), 913-930. doi: 10.3934/jimo.2017082. Google Scholar

[10]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No.1. Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7. Google Scholar

[11]

G. S. F. Frederico and D. F. M. Torres, Noether's symmetry theorem for variational and optimal control problems with time delay, Numerical Algebra, Control and Optimization, 2 (2012), 619-630. doi: 10.3934/naco.2012.2.619. Google Scholar

[12]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413. Google Scholar

[13]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365. doi: 10.1002/oca.843. Google Scholar

[14]

Z. H. GongC. 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

[15]

K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855. Google Scholar

[16]

C. Y. Kaya and J. L. Noakes, Computational method for time-optimal Switching Control, Journal of Optimization Theory and Applications, 117 (2003), 69-92. doi: 10.1023/A:1023600422807. Google Scholar

[17]

H. W. J. LeeK. L. TeoV. Rehbock and L. S. Jennings, Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407. doi: 10.1016/S0005-1098(99)00050-3. Google Scholar

[18]

H. W. J. Lee and K. H. Wong, Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints, Optimization Methods and Software, 21 (2006), 679-691. doi: 10.1080/10556780500142306. Google Scholar

[19]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar

[20]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, A new computational method for optimizing nonlinear impulsive systems, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 18 (2011), 59-76. doi: 20.500.11937/48414. Google Scholar

[21]

C. Y. LiuR. 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 and Z. H. Gong, Modelling and optimal control of a time-delayed switched system in fed-batch process, Journal of the Franklin Institute, 351 (2014), 840-856. doi: 10.1016/j.jfranklin.2013.09.014. Google Scholar

[23]

C. Y. LiuZ. H. GongK. L. TeoJ. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar

[24]

C. Y. LiuZ. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368. doi: 10.1016/j.apm.2017.09.007. Google Scholar

[25]

C. Y. LiuR. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7. Google Scholar

[26]

R. LoxtonQ. Lin and K. L. Teo, Switching time optimization for nonlinear switched systems: Direct optimization and the time-scaling transformation, Pacific Journal of Optimization, 10 (2014), 537-560. doi: 20.500.11937/19892. Google Scholar

[27]

R. LoxtonK. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2018), 2923-2929. doi: 10.1016/j.automatica.2008.04.011. Google Scholar

[28]

R. LoxtonK. L. Teo and V. Rehbock, An optimization approach to state-delay identification, IEEE Transactions on Automatic Control, 55 (2010), 2113-2119. doi: 10.1109/TAC.2010.2050710. Google Scholar

[29]

R. LoxtonK. L. TeoV. Rehbock and K. F. C. Yiu, Control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029. Google Scholar

[30]

R. LoxtonK. L. TeoV. Rehbock and W. K. Ling, Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031. Google Scholar

[31]

R. Luus, Use of Luus-Jaakola optimization procedure for singular optimal control problems, Nonlinear Analysis, Theory, Methods & Applications, 47 (2001), 5647-5658. doi: 10.1016/S0362-546X(01)00666-6. Google Scholar

[32]

H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM Journal on Control Optimization, 42 (2004), 2239-2263. doi: 10.1137/S0363012902402578. Google Scholar

[33]

J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer Series in Operations Research and Financial Engineering, New York, 2006. Google Scholar

[34]

M. SchlegelK. StockmannT. Binder and W. Marquardt, Dynamic optimization using adaptive control vector parameterization, Computers and Chemical Engineering, 29 (2005), 1731-1751. doi: 10.1016/j.compchemeng.2005.02.036. Google Scholar

[35]

A. Siburian and V. Rehbock, Numerical procedure for solving a class of singular optimal control problems, Optimization Methods and Software, 19 (2004), 413-426. doi: 10.1080/10556780310001656637. Google Scholar

[36]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. Google Scholar

[37]

L. Y. WangW. H. GuiK. L. TeoR. Loxton and C. H. Yang, Optimal control problems arising in the zinc sulphate electrolyte purification process, Journal of Global Optimization, 54 (2012), 307-323. doi: 10.1007/s10898-012-9863-x. Google Scholar

[38]

L. Y. WangW. H. GuiK. L. TeoR. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705. Google Scholar

[39]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972. Google Scholar

[40]

K. H. WongL. S. Jennings and F. Benyah, The control parametrization enhancing transform for constrained time–delayed optimal control problems, ANZIAM Journal, 43 (2002), 154-185. doi: 10.21914/anziamj.v43i0.469. Google Scholar

[41]

S. F. WoonV. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594. doi: 10.1002/oca.1015. Google Scholar

[42]

C. Z. Wu and K. L. Teo, Optimal impulsive control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. Google Scholar

[43]

C. Z. WuK. L. TeoR. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067. doi: 10.1016/j.aml.2005.11.018. Google Scholar

[44]

X. Xiang and Y. Peng, Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Management Optimization, 4 (2008), 17-32. doi: 10.3934/jimo.2008.4.17. Google Scholar

[45]

C. J. YuB. LiR. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7. Google Scholar

[46]

C. J. YuQ. LinR. LoxtonK. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, Journal of Optimization Theory and Applications, 169 (2016), 876-901. doi: 10.1007/s10957-015-0783-z. Google Scholar

[47]

C. J. YuK. L. TeoL. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar

[48]

Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693. doi: 10.1021/ie200996f. Google Scholar

Figure 1.  The relationship between $ (s, \mu(s|\mathit{\boldsymbol{\sigma}})) $ and $ (\omega(s|\mathit{\boldsymbol{\sigma}}), \mu(s|\mathit{\boldsymbol{\sigma}})-h) $. Note that $ \mu(s|\mathit{\boldsymbol{\sigma}}) $ is a piecewise linear function
Figure 2.  The relationship between $ (s, t) $ and $ (\omega(s), \mu(\omega(s))) $
Figure 3.  A demonstration for two scenarios of $ \sigma_{\lfloor s\rfloor +1} $
Figure 4.  Optimal state trajectory for Problem 1 for $ \sigma_i\in[0, 1) $
Figure 5.  Optimal state trajectory for Problem 2
Figure 6.  Optimal control $ \xi_1, \xi_2, \xi_3, \xi_4 $ for Problem 2
Table 1.  Optimal cost if the interval of $ \sigma_i $ is different
the interval of $ \sigma_i $ $ \tilde{{G}}^*_{0}(s, \mathit{\boldsymbol{\sigma}}) $ the optimal $ \mathit{\boldsymbol{\sigma}} $
$ \sigma_i\in[0, 1] $ $ 3.9855\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.000, 0.46836125, 0.53163875) $
$ \sigma_i\in[0.001, 1) $ $ 4.0432\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.001, 0.46779421, 0.53120579) $
$ \sigma_i\in[0.01, 1) $ $ 4.5411\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.010, 0.46257412, 0.52742588) $
the interval of $ \sigma_i $ $ \tilde{{G}}^*_{0}(s, \mathit{\boldsymbol{\sigma}}) $ the optimal $ \mathit{\boldsymbol{\sigma}} $
$ \sigma_i\in[0, 1] $ $ 3.9855\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.000, 0.46836125, 0.53163875) $
$ \sigma_i\in[0.001, 1) $ $ 4.0432\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.001, 0.46779421, 0.53120579) $
$ \sigma_i\in[0.01, 1) $ $ 4.5411\times 10^{-3} $ $ \mathit{\boldsymbol{\sigma}}^\star=(0.010, 0.46257412, 0.52742588) $
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