# American Institute of Mathematical Sciences

March  2022, 18(2): 1305-1320. doi: 10.3934/jimo.2021021

## On the optimal control problems with characteristic time control constraints

 Shanghai University, Shanghai, 200444, China

* Corresponding author: Shuxuan Su

Received  June 2020 Revised  December 2020 Published  March 2022 Early access  January 2021

Fund Project: This work is supported by National Natural Science Foundation of China(NSFC), Grant No.11871039 and Science and Technology Commission of Shanghai Municipality(STCSM), Grant No. 20JC1413900

In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples.

Citation: Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1305-1320. doi: 10.3934/jimo.2021021
##### References:

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##### References:
Time-scaling function for pre-fixed terminal time
Optimal control for Problem 1 with $p = 9$.
Optimal state trajectories for Problem 1 obtained using the two different methods with $p = 9$.
Optimal controls for Problem 2 with $p = 5$
Optimal controls for Problem 2 with $p = 6$
Optimal controls for Problem 2 with p = 7
Optimal state trajectories for Problem 2 with different partition numbers
Optimal costs for Problem 1 obtained using the two different methods
 Knots $G_0^*$ Traditional control parameterization New time-scaling transformation $p=5$ $110.8$ $1.9531$ $p=7$ $42.34$ $1.6671$ $p=9$ $21.79$ $1.6053$
 Knots $G_0^*$ Traditional control parameterization New time-scaling transformation $p=5$ $110.8$ $1.9531$ $p=7$ $42.34$ $1.6671$ $p=9$ $21.79$ $1.6053$
Optimal costs for Problem 2 using different lower bounds on each $\theta_l$ with different partition $p$
 Lower bounds on $\theta_l\ (l=1, \ldots, p)$ $J_0^*$ $p=5$ $p=6$ $p=7$ $\geq 0.05$ $12.189$ $8.113$ $4.572$ $\geq 0.1$ $12.347$ $8.258$ $4.673$ $\geq 0.5$ $13.147$ $9.442$ $5.506$ $\geq 1$ $14.453$ $10.924$ $7.000$
 Lower bounds on $\theta_l\ (l=1, \ldots, p)$ $J_0^*$ $p=5$ $p=6$ $p=7$ $\geq 0.05$ $12.189$ $8.113$ $4.572$ $\geq 0.1$ $12.347$ $8.258$ $4.673$ $\geq 0.5$ $13.147$ $9.442$ $5.506$ $\geq 1$ $14.453$ $10.924$ $7.000$
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