# American Institute of Mathematical Sciences

## 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, doi: 10.3934/dcdss.2020098
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##### 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
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