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A new numerical method is presented to solve optimal control problem of a chaotic system based on Gauss pseudospectral method (GPM). Firstly, the Lagrange interpolation polynomials are constructed on Legendre-Gauss nodes and used to parameterize the state and control the trajectories in optimal control of the chaotic system. Then, the chaotic optimal control problem in the continuous space is transformed into a nonlinear programming (NLP) problem through GPM. Furthermore, the NLP problem is solved by the sequential quadratic programming algorithm. Finally, the proposed method is applied to the optimal control of the typical Lorenz, Chen, and Liu chaotic systems respectively. The simulation processes indicate that the GPM is effective, fast and feasible for solving optimal control problems of chaotic systems.
[1] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[2] Bemardo M D 1996 Phys. Lett. A 214 7139
[3] Zhu S P, Qian F C, Liu D 2010 Acta Phys. Sin. 59 2250 (in Chinese) [朱少平, 钱富才, 刘丁 2010 物理学报 59 2250]
[4] Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 0731 (in Chinese) [王兴元, 王明军 2008 物理学报 57 0731]
[5] Liu D, Qian F C, Ren H P 2004 Acta Phys. Sin. 53 2074 (in Chinese) [刘丁, 钱富才, 任海鹏 2004 物理学报 53 2074]
[6] Zhang J S, Xiao X C 2001 Acta Phys. Sin. 50 2092 (in Chinese) [张家树, 肖先赐 2001 物理学报 50 2092]
[7] Tan W, Wang Y N 2004 Acta Phys. Sin. 53 4087 (in Chinese) [谭文, 王耀南 2004 物理学报 53 4087]
[8] Liu F C, Liang X M, Song J Q 2008 Acta Phys. Sin. 57 1458 (in Chinese) [刘福才, 梁晓明, 宋佳秋 2008 物理学报 57 1458]
[9] Chen L Q 2002 Chin. Phys. 11 900
[10] Zhang S H, Shen K 2003 Chin. Phys. 12 149
[11] Sun F Y 2006 Chin. Phys. Lett. 23 32
[12] Ma M, Zheng Y A, Hu F Y, Zhao L 2009 Computer Simulation 28 409 (in Chinese) [马明, 郑永爱, 胡冯仪, 赵磊 2009 计算机仿真 28 409]
[13] Rafikov M, Balthazar J M 2004 Phys. Lett. A 333 241
[14] Awad El-Gohary, Ammar Sarhan 2006 Chaos, Solitons and Fractals 30 1122
[15] Awad El-Gohary 2006 Chaos, Solitons and Fractals 27 345
[16] Liu W, Chen G 2003 Int. J. Bifurcation and chaos 13 261
[17] Betts J T 1998 Journal of Guidance, Control, and Dynamics 21 193
[18] Bryson A E, Ho Y C 1975 Applied Optimal Control: Optimization, Estimation and Control (New York: John Wiley and Sons)
[19] Kirk D E 2003 Optimal Control Theory: An Introduction (New York: Dover Publications)
[20] Benson D A 2005 Ph. D. Dissertation (Massachusetts Institute of Technology, USA)
[21] Rao A V, Benson D, Darby C, Patterson M A, Francolin C, Sanders I, Huntington G T 2010 ACM Trans. Mathematical Software 37 1
[22] Boggs P T, Jon W T 1995 J. Comp. Appl. Math. 124 123
[23] Philip E G 2005 SIAM Review 47 99
[24] Cao X Q, Song J Q, Zhang W M, Zhao J 2011 Acta Phys. Sin. 60 070511 (in Chinese) [曹小群, 宋君强, 张卫民, 赵军 2011 物理学报 60 070511]
[25] Cao X Q, Song J Q, Zhang W M, Zhu X Q 2011 Acta Phys. Sin. 60 080401 (in Chinese) [曹小群, 宋君强, 张卫民, 朱小谦 2011 物理学报 60 080401]
[26] He J H 2008 Int. J. Modern. Phys. B 22 3487
[27] He J H 2000 Appl. Math. Mech. 21 797
[28] He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309
[29] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[30] Peng H J, Gao Q, Wu Z G, Zhong W X 2011 Acta Auto. Sin. 37 1248 (in Chinese) [彭海军, 高强, 吴志刚, 钟万勰 2011 自动化学报 37 1248]
[31] Peng H J, Gao Q, Wu Z G, Zhong W X 2010 Appl. Math. Mech. 31 1251
[32] Gao Q, Peng H J, Wu Z G, Zhong W X 2010 Journal of Dynamics and Control 8 1 (in Chinese) [高强, 彭海军, 吴志刚, 钟万勰 2010 动力学与控制学报 8 1]
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[1] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[2] Bemardo M D 1996 Phys. Lett. A 214 7139
[3] Zhu S P, Qian F C, Liu D 2010 Acta Phys. Sin. 59 2250 (in Chinese) [朱少平, 钱富才, 刘丁 2010 物理学报 59 2250]
[4] Wang X Y, Wang M J 2008 Acta Phys. Sin. 57 0731 (in Chinese) [王兴元, 王明军 2008 物理学报 57 0731]
[5] Liu D, Qian F C, Ren H P 2004 Acta Phys. Sin. 53 2074 (in Chinese) [刘丁, 钱富才, 任海鹏 2004 物理学报 53 2074]
[6] Zhang J S, Xiao X C 2001 Acta Phys. Sin. 50 2092 (in Chinese) [张家树, 肖先赐 2001 物理学报 50 2092]
[7] Tan W, Wang Y N 2004 Acta Phys. Sin. 53 4087 (in Chinese) [谭文, 王耀南 2004 物理学报 53 4087]
[8] Liu F C, Liang X M, Song J Q 2008 Acta Phys. Sin. 57 1458 (in Chinese) [刘福才, 梁晓明, 宋佳秋 2008 物理学报 57 1458]
[9] Chen L Q 2002 Chin. Phys. 11 900
[10] Zhang S H, Shen K 2003 Chin. Phys. 12 149
[11] Sun F Y 2006 Chin. Phys. Lett. 23 32
[12] Ma M, Zheng Y A, Hu F Y, Zhao L 2009 Computer Simulation 28 409 (in Chinese) [马明, 郑永爱, 胡冯仪, 赵磊 2009 计算机仿真 28 409]
[13] Rafikov M, Balthazar J M 2004 Phys. Lett. A 333 241
[14] Awad El-Gohary, Ammar Sarhan 2006 Chaos, Solitons and Fractals 30 1122
[15] Awad El-Gohary 2006 Chaos, Solitons and Fractals 27 345
[16] Liu W, Chen G 2003 Int. J. Bifurcation and chaos 13 261
[17] Betts J T 1998 Journal of Guidance, Control, and Dynamics 21 193
[18] Bryson A E, Ho Y C 1975 Applied Optimal Control: Optimization, Estimation and Control (New York: John Wiley and Sons)
[19] Kirk D E 2003 Optimal Control Theory: An Introduction (New York: Dover Publications)
[20] Benson D A 2005 Ph. D. Dissertation (Massachusetts Institute of Technology, USA)
[21] Rao A V, Benson D, Darby C, Patterson M A, Francolin C, Sanders I, Huntington G T 2010 ACM Trans. Mathematical Software 37 1
[22] Boggs P T, Jon W T 1995 J. Comp. Appl. Math. 124 123
[23] Philip E G 2005 SIAM Review 47 99
[24] Cao X Q, Song J Q, Zhang W M, Zhao J 2011 Acta Phys. Sin. 60 070511 (in Chinese) [曹小群, 宋君强, 张卫民, 赵军 2011 物理学报 60 070511]
[25] Cao X Q, Song J Q, Zhang W M, Zhu X Q 2011 Acta Phys. Sin. 60 080401 (in Chinese) [曹小群, 宋君强, 张卫民, 朱小谦 2011 物理学报 60 080401]
[26] He J H 2008 Int. J. Modern. Phys. B 22 3487
[27] He J H 2000 Appl. Math. Mech. 21 797
[28] He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309
[29] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[30] Peng H J, Gao Q, Wu Z G, Zhong W X 2011 Acta Auto. Sin. 37 1248 (in Chinese) [彭海军, 高强, 吴志刚, 钟万勰 2011 自动化学报 37 1248]
[31] Peng H J, Gao Q, Wu Z G, Zhong W X 2010 Appl. Math. Mech. 31 1251
[32] Gao Q, Peng H J, Wu Z G, Zhong W X 2010 Journal of Dynamics and Control 8 1 (in Chinese) [高强, 彭海军, 吴志刚, 钟万勰 2010 动力学与控制学报 8 1]
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