A Monotonic Time-Discretized Scheme for Optimal Control Problems

XIANG Qingqing; CHEN Hao

doi:10.21656/1000-0887.460020

Volume 47 Issue 2

Feb. 2026

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XIANG Qingqing, CHEN Hao. A Monotonic Time-Discretized Scheme for Optimal Control Problems[J]. Applied Mathematics and Mechanics, 2026, 47(2): 230-242. doi: 10.21656/1000-0887.460020

Citation:

XIANG Qingqing, CHEN Hao. A Monotonic Time-Discretized Scheme for Optimal Control Problems[J]. Applied Mathematics and Mechanics, 2026, 47(2): 230-242. doi: 10.21656/1000-0887.460020

Citation:

XIANG Qingqing, CHEN Hao. A Monotonic Time-Discretized Scheme for Optimal Control Problems[J]. Applied Mathematics and Mechanics, 2026, 47(2): 230-242. doi: 10.21656/1000-0887.460020

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A Monotonic Time-Discretized Scheme for Optimal Control Problems

doi: 10.21656/1000-0887.460020

XIANG Qingqing,
CHEN Hao^,

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, P.R.China

Received Date: 2025-02-06
Rev Recd Date: 2025-11-07
Publish Date: 2026-02-01

Abstract

Abstract

Recently, Breitenbach and Borzì proposed a sequential quadratic Hamiltonian method for solving optimal control problems. They proved the monotonic convergence of the algorithm in the continuous time case. However, the properties of the discrete version of the iterative procedure have not been tackled yet. A midpoint time-discretized scheme preserving the monotonic properties of the sequential quadratic Hamiltonian method was presented. Numerical experiments show the effectiveness and convergence of the proposed algorithm.
- optimal control problem,
- sequential quadratic Hamiltonian method,
- monotonic time-discretized scheme,
- convergence

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References(16)

References

[1]	郑明亮. 机械多体系统动力学非线性最优控制问题的Noether理论[J]. 应用数学和力学, 2018, 39(7): 776-784. doi: 10.21656/1000-0887.380295 ZHENG Mingliang. The Noether theorem for nonlinear optimal control problems of mechanical multibody system dynamics[J]. Applied Mathematics and Mechanics, 2018, 39(7): 776-784. (in Chinese) doi: 10.21656/1000-0887.380295
[2]	HULL D G. Optimal Control Theory for Applications[M]. New York: Springer, 2003.
[3]	BRYSON A E, HO Y C, SIOURIS G M. Appliedoptimal control: optimization, estimation, and control[J]. IEEE Transactions on Systems, Man, and Cybernetics, 1979, 9(6): 366-367. doi: 10.1109/TSMC.1979.4310229
[4]	KAMIEN M I, SCHWARTZ N L. Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management[M]. North Holland Inc, 1998.
[5]	程国采. 航天飞行器最优控制理论与方法[M]. 北京: 国防工业出版社, 1999. CHENG Guocai. Spacecraft Optimal Control Theory and Method[M]. Beijing: National Defense Industry Press, 1999. (in Chinese)
[6]	MOHAMMADI S, HEJAZI S. Using particle swarm optimization and genetic algorithms for optimal control of non-linear fractional-order chaotic system of cancer cells[J]. Mathematics and Computers in Simulation, 2022, 206: 538-560.
[7]	王昕炜, 彭海军, 钟万勰. 具有潜伏期时滞的时变SEIR模型的最优疫苗接种策略[J]. 应用数学和力学, 2019, 40(7): 701-712. doi: 10.21656/1000-0887.400048 WANG Xinwei, PENG Haijun, ZHONG Wanxie. Optimal vaccination strategies for a time-varying SEIR epidemic model with latent delay[J]. Applied Mathematics and Mechanics, 2019, 40(7): 701-712. (in Chinese) doi: 10.21656/1000-0887.400048
[8]	PONTRYAGIN L S. The Mathematical Theory of Optimal Processes[M]. New York: Interscience Publishers, 1962.
[9]	BELLMAN R E, DREYFUS S E. Applied Dynamic Programming[M]. Princeton: Princeton University Press, 1962.
[10]	陈学松, 刘富春. 一类非线性动态系统基于强化学习的最优控制[J]. 控制与决策, 2013, 28(12): 1889-1893. CHEN Xuesong, LIU Fuchun. Optimal control for a class of nonlinear dynamic systems based on reinforcement learning[J]. Control and Decision, 2013, 28(12): 1889-1893. (in Chinese)
[11]	APEL T, FLAIG T G. Crank: Nicolson schemes for optimal control problems with evolution equations[J]. SIAM Journal on Numerical Analysis, 2012, 50(3): 1484-1512. doi: 10.1137/100819333
[12]	LIU X, FRANK J. Symplectic Runge-Kutta discretization of a regularized forward-backward sweep iteration for optimal control problems[J]. Journal of Computational and Applied Mathematics, 2021, 383: 113133. doi: 10.1016/j.cam.2020.113133
[13]	ZHANG Z, CHEN X. A global convergent semi-smooth Newton method for semilinear elliptic optimal control problem[J]. Computers and Mathematics With Applications, 2022, 120: 13.
[14]	ZHANG Z, CHEN X. A conjugate gradient method for distributed optimal control problems with nonhomogeneous Helmholtz equation[J]. Applied Mathematics and Computation, 2021, 402: 126019. doi: 10.1016/j.amc.2021.126019
[15]	BORZÌ A. The Sequential Quadratic Hamiltonian Method: Solving Optimal Control Problems[M]. CRC Press, 2023.
[16]	BREITENBACH T, BORZÌ A. A sequential quadratic Hamiltonian scheme for solving non-smooth quantum control problems with sparsity[J]. Journal of Computational and Applied Mathematics, 2020, 369: 112583. doi: 10.1016/j.cam.2019.112583