# American Institute of Mathematical Sciences

June  2020, 13(6): 1683-1695. doi: 10.3934/dcdss.2020098

## Time-scaling transformation for optimal control problem with time-varying delay

 1 Department of Mathematics, Shanghai University, Shanghai 200444, China 2 School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China

* Corresponding author: Yanqin Bai

Received  February 2018 Revised  May 2018 Published  September 2019

Fund Project: This research was supported by grants from the National Natural Science Foundation of China (No.11771275, No.11871039)

This paper focuses on the solution to nonlinear time-delay optimal control problems with time-varying delay subject to canonical equality and inequality constraints. Traditional control parameterization in conjunction with time-scaling transformation could optimize control parameters and switching times at the same time when the time-delays in the dynamic system are taken as constants. The purpose of this paper is to extend this method to solve the dynamic systems with time-varying delay. We introduce a hybrid time-scaling transformation that converts the given time-delay system into an equivalent system defined on a new time horizon with fixed switching times. Meanwhile, we obtain the value of time-delay state utilizing the relationship between the new time scale and the original one. After computing the gradients of the cost and constraints with respect to the control heights and its durations, we could solve the equivalent optimal control problem using gradient based optimization method.

Citation: Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098
##### References:

show all references

##### References:
Optimal control for Problem 1 for the case of $q = 7$
Optimal state trajectory for Problem 1 using hybrid time-scaling transformation for the case of $q = 7$
Optimal state trajectory for Problem 1 using traditional control parameterization with fixed switching times ($q = 7$)
Optimal control for Problem 2 for the case of $q = 8$
Optimal state trajectory for Problem 2 using hybrid time-scaling transformation for the case of $q = 8$
Optimal state trajectory for Problem 2 using traditional control parameterization with fixed switching times ($q = 8$)
Optimal cost of Problem 1
 (a) using hybrid time-scaling transformation (b) using traditional control parameterization partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ $q=7$ 2.1275 $q=7$ 2.1347 $q=5$ 2.1333 $q=5$ 2.1446 $q=3$ 2.1340 $q=3$ 2.1812
 (a) using hybrid time-scaling transformation (b) using traditional control parameterization partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ $q=7$ 2.1275 $q=7$ 2.1347 $q=5$ 2.1333 $q=5$ 2.1446 $q=3$ 2.1340 $q=3$ 2.1812
Parameters in Problem 2
 a b c $t_{f}$ Q R S 0.2 0.5 0.2 1.5 $I_{2\times2}$ $I_{2\times2}$ $10^{4}I_{2\times2}$
 a b c $t_{f}$ Q R S 0.2 0.5 0.2 1.5 $I_{2\times2}$ $I_{2\times2}$ $10^{4}I_{2\times2}$
Optimal cost of Problem 2
 (a) Using hybrid time-scaling transformation (b) Using traditional control parameterization partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ $q=8$ 3.8750 $q=8$ 5.3252 $q=6$ 4.2319 $q=6$ 6.1942 $q=4$ 6.9578 $q=4$ 7.9119
 (a) Using hybrid time-scaling transformation (b) Using traditional control parameterization partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ partition number $g_{0}(\mathit{\boldsymbol{u}}^{q,*})$ $q=8$ 3.8750 $q=8$ 5.3252 $q=6$ 4.2319 $q=6$ 6.1942 $q=4$ 6.9578 $q=4$ 7.9119
 [1] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 [2] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113 [3] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [4] Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 [5] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 [6] Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 [7] Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 [8] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [9] Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 [10] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [11] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [12] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [13] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [14] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [15] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [16] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [17] Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166 [18] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [19] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [20] Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

2019 Impact Factor: 1.233