Abstract
In some recent work on nonlinear filtering, current algebras play a significant role. In the study of certain control problems arising in spacecraft attitude control, it has also become apparent that Lie-Poisson algebras arise naturally. It is our purpose in this paper to draw attention to some of the mathematical and system-theoretic aspects of these two types of infinite-dimensional Lie algebras. To this end, this paper contain brief background material on the structure of these Lie algebras. Part I of this paper deals mainly with current algebras and Part II of this paper deals with Lie-Poisson algebras. In both parts of the paper we indicate the specific applications that we are most familiar with. We then proceed to suggest other types of application areas where the structure of these infinite-dimensional Lie algebras may be exploited. Specifically we have in mind the control of infinite dimensional bilinear systems and complex multi-body mechanical systems. Details of the existing applications may be found in the references.
Partial support for this work was provided by the Department of Energy under Contract DEAC01-80-RA50420-A001 and by National Science Foundation under Grant ECS-81-18138.
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References
R. Abraham and J. Marsden (1978), Foundations of Mechanics, 2nd edition, Addison-Wesley, Reading, MA.
V.I. Arnold (1978), Mathematical Methods of Classical Mechanics, Springer Graduate Texts in Math, No. 60, Springer-Verlag, New York.
J. Baillieul (1978), Geometric methods for nonlinear optimal control problems, J. Optimization Theory and Applications, vol. 25, no. 4, 519–548.
D. Ebin (1970). The manifold of Riemannian metrics. In Proc. Symp. Pure Math., XI, American Math. Soc., Providence.
J. Eels (1966). A setting for global analysis. Bulletin of the American Math. Soc., 72, 751–807.
M. El-Sayed and P.S. Krishnaprasad (1981), Homogeneous interconnected systems: an example, IEEE Transactions on Automatic Control, vol. AC-26, no. 4, 894–901.
V. Guillemin and S. Sternberg (1977), Geometric Asymptotics, Am. Math. Soc. Survey, Vol. 14, American Math. Society, Providence, RI.
R. Hermann (1978), Toda Lattices, Cosymplectic Manifolds, Bäcklund Transformations and Kinks Part A, Math. Sci. Press, Boston.
P. Holmes and J. Marsden (1983), Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana University Mathematics Journal, 32, No. 2, pp. 273–309.
A.A. Kirillov (1974), Elements of the Theory of Representations, Springer-Verlag, New York.
B. Kostant (1979), Quantization and representation theory, in M.F. Atiyah, ed. Representation Theory of Lie Groups, Proc. SRC/LMS Research symposium on Lie Groups, Oxford 1977, Cambridge University Press, Cambridge.
P.S. Krishnaprasad, S.I. Marcus and M. Hazewinkel (1983), Current Algebras and the identification problem, Stochastics, (to appear).
P.S. Krishnaprasad (1983), Lie-Poisson structures and dual-spin spacecraft, Dept. of Electrical Engg. Tech. Research Report, SRR 83-4, University of Maryland. Also Proc 22nd IEEE Conf. on Decision and Control.
P.S. Krishnaprasad (1983), Lie-Poisson structures and dual-spin spacecraft, II: Asymptotic behavior, Dept. of Electrical Engg. Tech. Research Report, SRR 83-7.
P.S. Krishnaprasad and C.A. Berenstein (1983), On the equilibria of rigid spacecraft with rotors, submitted to Systems and Control Letters, SRR 83-6.
B.A. Kupershmidt and Yu. Manin (1977), Equations of long waves with free surface, Functional Analysis and its Applications, 11, pp. 188–197.
P.K. Mitter (1980). Geometry of the space of gauge orbits and the Yang-Mills dynamical system. In G.T. 'Hooft et. al. (ed.) Recent Developments in Gauge Theories (Cargese School), Plenum Press, New York.
P.K. Mitter and C.M. Viallet (1981). On the bundle of connections and the gauge orbit manifold in Yang-Mills theory. Commun. Math. Phys., 79, 457–472.
J. Marsden (1974). Application of Global Analysis in Mathematical Physics, Publish or Perish, Boston.
M.S. Narasimhan and T.R. Ramadas (1979). Geometry of SU(2) gauge fields. Commu. Math. Phys., 67, 121–136.
R.S. Palais (1965). Seminar on the Atiyah-Singer INdex Theorem, Annals. of Mathematics Studies No. 57, Princeton University Press, Princeton.
R.S. Palais (1968). Foundations of Global Nonlinear Analysis, W.A. Benjamin, New York.
T. Ratiu (1980), The Motion of the free n-dimensional rigid body, Indiana Univesity Math. J., 29, No. 4, pp. 609–629.
A.G. Reyman and M.A. Semenov-Tian-Shansky (1981), Reduction of Hamiltonian Systems, affine Lie algebras and Lax equations II, Invent. Math., 63, pp. 423–432.
A.G. Reyman and M.A. Semenov-Tian-Shansky (1980), Current algebras and nonlinear partial differential equations, Soviet Math. Dokl, 21, No. 2, pp. 630–634.
S. Steinberg (1977), Applications of the Lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations. Journal of Differential Equations, 26, 404–434.
J. Wei and E. Norman (1964). On the global representation of solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc., April, 327–334.
J. Wei and E. Norman (1963). On the Lie algebraic solution of linear differential equations. J. Math. Phys., 4, No. 4, 575–581.
A. Weinstein (1983), The local structure of Poisson manifolds, J. Diff. Geometry (to appear).
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Krishnaprasad, P.S. (1984). On certain infinite-dimensional lie algebras and related system-theoretic problems. In: Fuhrmann, P.A. (eds) Mathematical Theory of Networks and Systems. Lecture Notes in Control and Information Sciences, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0031085
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DOI: https://doi.org/10.1007/BFb0031085
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