## Abstract and Applied Analysis

### A Simple Exact Penalty Function Method for Optimal Control Problem with Continuous Inequality Constraints

#### Abstract

We consider an optimal control problem subject to the terminal state equality constraint and continuous inequality constraints on the control and the state. By using the control parametrization method used in conjunction with a time scaling transform, the constrained optimal control problem is approximated by an optimal parameter selection problem with the terminal state equality constraint and continuous inequality constraints on the control and the state. On this basis, a simple exact penalty function method is used to transform the constrained optimal parameter selection problem into a sequence of approximate unconstrained optimal control problems. It is shown that, if the penalty parameter is sufficiently large, the locally optimal solutions of these approximate unconstrained optimal control problems converge to the solution of the original optimal control problem. Finally, numerical simulations on two examples demonstrate the effectiveness of the proposed method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 752854, 12 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412276979

Digital Object Identifier
doi:10.1155/2014/752854

Mathematical Reviews number (MathSciNet)
MR3208563

Zentralblatt MATH identifier
07023020

#### Citation

Gao, Xiangyu; Zhang, Xian; Wang, Yantao. A Simple Exact Penalty Function Method for Optimal Control Problem with Continuous Inequality Constraints. Abstr. Appl. Anal. 2014 (2014), Article ID 752854, 12 pages. doi:10.1155/2014/752854. https://projecteuclid.org/euclid.aaa/1412276979

#### References

• Y. Sakawa and Y. Shindo, “Optimal control of container of container cranes,” Automatica, vol. 18, no. 3, pp. 257–266, 1982.
• C. H. Jiang, Q. Lin, C. Yu, K. L. Teo, and G.-R. Duan, “An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 30–53, 2012.
• X. L. Liu and G. R. Duan, “Nonlinear optimal control for the soft landing of lunar lander,” in Proceedings of the 1st International Symposium on Systems and Control in Aerospace and Astronautics, pp. 1382–1387, Harbin, China, January 2006.
• X. Y. Gao and K. L. Teo, “Fuel optimal control of nonlinear spacecraft rendezvous system with collision avoidance constraint,” submitted to IEEE Transactions on Automatic Control.
• C. Büskens and H. Maurer, “SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control,” Journal of Computational and Applied Mathematics, vol. 120, no. 1-2, pp. 85–108, 2000.
• W. Huyer and A. Neumaier, “A new exact penalty function,” SIAM Journal on Optimization, vol. 13, no. 4, pp. 1141–1158, 2003.
• T. W. Chen and V. S. Vassiliadis, “Inequality path constraints in optimal control: a finite iteration $\varepsilon$-convergent scheme based on pointwise discretization,” Journal of Process Control, vol. 15, no. 3, pp. 353–362, 2005.
• M. Gerdts, “Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems,” SIAM Journal on Optimization, vol. 19, no. 1, pp. 326–350, 2008.
• M. Gerdts, “A nonsmooth Newton's method for control-state constrained optimal control problems,” Mathematics and Computers in Simulation, vol. 79, no. 4, pp. 925–936, 2008.
• J. Chen and M. Gerdts, “Numerical solution of control-state constrained optimal control problems with an inexact smoothing Newton method,” IMA Journal of Numerical Analysis, vol. 31, no. 4, pp. 1598–1624, 2011.
• K. L. Teo, L. S. Jennings, H. W. J. Lee, and V. Rehbock, “The control parameterization enhancing transform for constrained optimal control problems,” Journal of the Australian Mathematical Society B: Applied Mathematics, vol. 40, no. 3, pp. 314–335, 1999.
• K. L. Teo and L. S. Jennings, “Nonlinear optimal control problems with continuous state inequality constraints,” Journal of Optimization Theory and Applications, vol. 63, no. 1, pp. 1–22, 1989.
• R. C. Loxton, K. L. Teo, V. Rehbock, and K. F. C. Yiu, “Optimal control problems with a continuous inequality constraint on the state and the control,” Automatica, vol. 45, no. 10, pp. 2250–2257, 2009.
• M. Gerdts and M. Kunkel, “A nonsmooth Newton's method for discretized optimal control problems with state and control constraints,” Journal of Industrial and Management Optimization, vol. 4, no. 2, pp. 247–270, 2008.
• K. L. Teo, C. J. Goh, and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, vol. 55, Longman, New York, NY, USA, 1991.
• B. Li, C. J. Yu, K. L. Teo, and G. R. Duan, “An exact penalty function method for continuous inequality constrained optimal control problem,” Journal of Optimization Theory and Applications, vol. 151, no. 2, pp. 260–291, 2011. \endinput