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Optimal Control, Stability and Numerical Integration on SU(3)

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Abstract

This work is a study on left-invariant, drift-free optimal control problem on SU(3) Lie group. The control objective is to minimize a cost function and satisfy the given dynamical constraints. The stability analysis of the resulting restricted dynamics is detailed. Also, numerical integration via two unconventional methods, Kahan’s integrator and Lie–Trotter integrator, is performed. A comparison between these two methods along with the conventional 4th-step Runge–Kutta method is thus presented.

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References

  1. Brockett, R.W.: System theory on group manifold and coset spaces. SIAM J. Control 10, 265–284 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jurdjevic, V., Sussmann, H.J.: Control systems on Lie groups. J. Differ. Equ. 12, 313–329 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Remsing, C.: Control and integrability on SO(3). Lecture Notes in Engineering and Computer Science, pp. 1705–1710 (2010)

  4. Puta, M., Birtea, P., Lǎzureanu, C., Pop, C., Tudoran, R.: Control, integrability and stability in some concrete mechanical problems on matrix Lie groups. Univ di Roma La Sapienza, Quad. Sem. Top. Alg. e Diff. (1998)

  5. Spindler, K.: Optimal control on Lie groups: theory and applications. WSEAS Trans. Math. 12, 531–542 (2013)

    Google Scholar 

  6. Pop, C.: An optimal control problem on the Heisenberg Lie group H(3). Gen. Math. 5, 323–330 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Lǎzureanu, C., Bînzar, T.: On a Hamiltonian version of controls dynamic for a drift-free left invariant control system on G4. In: International Conference of Differential Geometryand Dynamical Systems, vol. 9, Article ID 1250065 (2012)

  8. Craioveanu, M., Pop, C., Aron, A., Petrişor, C.: An optimal control problem on the special Euclidean group SE(3,\({\mathbb{R}}\)). In: The International Conference of Differential Geometryand Dynamical Systems, pp. 68–78 (2009)

  9. Wybourne, B.G.: Classical groups for physicists. Wiley, London (1974)

  10. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  11. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  12. Georgi, H.: Lie Algebras in Particle Physics: From Isospin to Unified Theories. Westview Press, Boulder (1999)

    MATH  Google Scholar 

  13. Leonard, N.E.: Averaging and motion control systems on Lie groups, Ph.D. Thesis, University of Maryland, College Park (1994)

  14. Krishnaprasad, P.S.: Optimal control and poisson reduction, Technical Report, T.R.93-87 (1993)

  15. Jay, L.O.: Preserving Poisson structure and orthogonality in numerical integration of differential equations. Comput. Math. Appl. 48, 237–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kahan, W.: Unconventional Numerical Methods for Trajectory Calculation, Lecture Notes (1993)

  17. Hirota, R., Kimura, K.: Discretization of Euler top. J. Phys. Soc. Jpn. 69, 627–630 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kimura, K., Hirota, R.: Discretization of Lagrange top. J. Phys. Soc. Jpn. 69, 3193–3199 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Petrera, M., Pfadler, A., Suris, Y.B.: On integrability of Hirota–Kimura type discretizations. Regul. Chaotic Dyn. 16, 245–289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Ceramic Engineering, National Institute of Technology, Rourkela, Odisha, India

    Soumya Prakash Sahoo

  2. Department of Mathematics, National Institute of Technology, Rourkela, Odisha, India

    Amit Jena & K. C. Pati

  3. Department of Chemical Engineering, National Institute of Technology, Rourkela, Odisha, India

    Soumya Ranjan Sahoo

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  1. Soumya Prakash Sahoo
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  2. Amit Jena
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  3. Soumya Ranjan Sahoo
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  4. K. C. Pati
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Correspondence to K. C. Pati.

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Sahoo, S.P., Jena, A., Sahoo, S.R. et al. Optimal Control, Stability and Numerical Integration on SU(3). Int. J. Appl. Comput. Math 3, 1661–1675 (2017). https://doi.org/10.1007/s40819-016-0181-8

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  • DOI: https://doi.org/10.1007/s40819-016-0181-8

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