# American Institute of Mathematical Sciences

December 2017, 7(4): 481-492. doi: 10.3934/naco.2017030

## Numerical method for solving optimal control problems with phase constraints

 Matrosov Institute for System Dynamics and Control Theory SB RAS, Lermontov str., 134,664033, Russia

* Corresponding author: tz@icc.ru

The authors are supported by RFBR grant 17-07-00627

Received  February 2017 Revised  September 2017 Published  October 2017

Fund Project: This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey

The main idea of the method consists in successive solving auxiliary problems, which minimizes a special constructed Lagrange function, subject to linearized phase constraints. The linearly constrained auxiliary problems are more simple than the original ones because linear constraints can be easily processed. We shall discuss different aspects connected with approximating control problems and using the program system for solving them. We shall then pay attention to optimal control problems with constraints on inertia of control functions. For illustrations, four control problems will be solved using the proposed software.

Citation: Alexander Tyatyushkin, Tatiana Zarodnyuk. Numerical method for solving optimal control problems with phase constraints. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 481-492. doi: 10.3934/naco.2017030
##### References:
 [1] A. Bondarenko, D. Bortz and J. More, A collection of large-scale nonlineary constrained optimization test problems, Optimization Online, 20 (1998), 18-32. [2] Yu. G. Evtushenko, Methods for Solving Extreme Problems and Their Application in Optimization Systems, Moscow, Nauka, 1982. (In Russian) [3] R. Gabasov, F. M. Kirillova and A. I. Tyatyushkin, Constructive Methods of Optimization. P. 1: Linear Problems, Minsk, University, 1984. [4] P. Gill, W. Murray and M. Wright, Practical Optimization, Moscow, Mir, 1985. [5] A. Yu. Gornov, The Computational Technologies for Solving Optimal Control Problems, Nauka, Novosibirsk, 2009. [6] A. Yu. Gornov, A. I. Tyatyushkin and E. A. Finkelstein, Numerical methods for solving terminal optimal control problems, Computational Mathematics and Mathematical Physics, 56 (2016), 221-234. doi: 10.1134/S0965542516020093. [7] A. Yu. Gornov and T. S. Zarodnyuk, Tunneling algorithm for solving nonconvex optimal control problems, Optimization, Simulation, and Control, Springer Optimization and Its Applications, 76 (2013), 289-299. [8] A. Yu. Gornov, T. S. Zarodnyuk, T. I. Madzhara, A. V. Daneyeva and I. A. Veyalko, A collection of test multiextremal optimal control problems, Optimization, Simulation, and Control, Springer Optimization and Its Applications, 76 (2013), 257-274. [9] V. I. Gurman, V. A. Baturin and I. V. Rasina, Approximate Methods of Optimal Control, Irkutsk, IGU Padlisher, 1983. (In Russian) [10] I. L. Junkins and I. D. Turner, Optimal continuous torque attitude maneuvers, Proc. AIAA/AAS Astrodynamics Conference, Palo Alto, Calif, 20 (1978), 78. [11] A. I. Tyatyushkin, A multimethod technique for solving optimal control problem, Optimization Letters, 7 (2012), 1335-1347. doi: 10.1007/s11590-011-0408-x. [12] A. I. Tyatyushkin and O. V. Morzhin, Constructive methods of control optimization in nonlinear systems, Automation and Remote Control, 70 (2009), 772-786. doi: 10.1134/S0005117909050063. [13] A. I. Tyatyushkin and O. V. Morzhin, On optimization of position control in attainability tube in a model problem, Journal of Computer and Systems Sciences International, 49 (2010), 740-749. doi: 10.1134/S1064230710050084. [14] A. I. Tyatyushkin and O. V. Morzhin, Numerical investigation of attainability sets of nonlinear controlled differential systems, Automation and Remote Control, 72 (2011), 1291-1300. doi: 10.1134/S0005117911060178. [15] Y. Wang, C. Yu and K. L. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control and Optimization, 6 (2013), 339-364. doi: 10.3934/naco.2016016. [16] A. I. Zholudev, A. I. Tyatyushkin and N. M. Erinchek, Numerical methods for optimization of control systems, Izvestiya: Technical cybernetics, 4 (1989), 18-32.

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##### References:
 [1] A. Bondarenko, D. Bortz and J. More, A collection of large-scale nonlineary constrained optimization test problems, Optimization Online, 20 (1998), 18-32. [2] Yu. G. Evtushenko, Methods for Solving Extreme Problems and Their Application in Optimization Systems, Moscow, Nauka, 1982. (In Russian) [3] R. Gabasov, F. M. Kirillova and A. I. Tyatyushkin, Constructive Methods of Optimization. P. 1: Linear Problems, Minsk, University, 1984. [4] P. Gill, W. Murray and M. Wright, Practical Optimization, Moscow, Mir, 1985. [5] A. Yu. Gornov, The Computational Technologies for Solving Optimal Control Problems, Nauka, Novosibirsk, 2009. [6] A. Yu. Gornov, A. I. Tyatyushkin and E. A. Finkelstein, Numerical methods for solving terminal optimal control problems, Computational Mathematics and Mathematical Physics, 56 (2016), 221-234. doi: 10.1134/S0965542516020093. [7] A. Yu. Gornov and T. S. Zarodnyuk, Tunneling algorithm for solving nonconvex optimal control problems, Optimization, Simulation, and Control, Springer Optimization and Its Applications, 76 (2013), 289-299. [8] A. Yu. Gornov, T. S. Zarodnyuk, T. I. Madzhara, A. V. Daneyeva and I. A. Veyalko, A collection of test multiextremal optimal control problems, Optimization, Simulation, and Control, Springer Optimization and Its Applications, 76 (2013), 257-274. [9] V. I. Gurman, V. A. Baturin and I. V. Rasina, Approximate Methods of Optimal Control, Irkutsk, IGU Padlisher, 1983. (In Russian) [10] I. L. Junkins and I. D. Turner, Optimal continuous torque attitude maneuvers, Proc. AIAA/AAS Astrodynamics Conference, Palo Alto, Calif, 20 (1978), 78. [11] A. I. Tyatyushkin, A multimethod technique for solving optimal control problem, Optimization Letters, 7 (2012), 1335-1347. doi: 10.1007/s11590-011-0408-x. [12] A. I. Tyatyushkin and O. V. Morzhin, Constructive methods of control optimization in nonlinear systems, Automation and Remote Control, 70 (2009), 772-786. doi: 10.1134/S0005117909050063. [13] A. I. Tyatyushkin and O. V. Morzhin, On optimization of position control in attainability tube in a model problem, Journal of Computer and Systems Sciences International, 49 (2010), 740-749. doi: 10.1134/S1064230710050084. [14] A. I. Tyatyushkin and O. V. Morzhin, Numerical investigation of attainability sets of nonlinear controlled differential systems, Automation and Remote Control, 72 (2011), 1291-1300. doi: 10.1134/S0005117911060178. [15] Y. Wang, C. Yu and K. L. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control and Optimization, 6 (2013), 339-364. doi: 10.3934/naco.2016016. [16] A. I. Zholudev, A. I. Tyatyushkin and N. M. Erinchek, Numerical methods for optimization of control systems, Izvestiya: Technical cybernetics, 4 (1989), 18-32.
The optimal control and trajectories for the problem 1
The optimal control and trajectories for the problem 2
The optimal control and trajectories for the problem 3
The optimal control and trajectories for the problem 4
The results of solving test problem Non-inertial Robot Arm
 Software $N = 10$ $N = 50$ $N = 100$ $N = 500$ LANCELOT -(0.1) -(16) -(140) - MINOS 9.278630 (0.2) 9.145749 (3.5) 9.141995 (110) 9.141334 (305) SNOPT 9.278630 (2.30) 9.145749 (64) -(10) -(315) LOQO -(14) -(154) -(194) - OPTCON 9.278615 (20) 9.147535 (37) 9.152146 (87) 9.148295 (309)
 Software $N = 10$ $N = 50$ $N = 100$ $N = 500$ LANCELOT -(0.1) -(16) -(140) - MINOS 9.278630 (0.2) 9.145749 (3.5) 9.141995 (110) 9.141334 (305) SNOPT 9.278630 (2.30) 9.145749 (64) -(10) -(315) LOQO -(14) -(154) -(194) - OPTCON 9.278615 (20) 9.147535 (37) 9.152146 (87) 9.148295 (309)
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