Abstract
The theory of differential equations and control have been linked very closely because most of the early applications of control theory were to engineering problems of the type which are most naturally described by ordinary differential equations. The questions of importance in control have helped to revitalize certain problem areas in differential equations and methods and tools from control have been useful in obtaining new results in differential equation theory. On the other hand, going back to the era of Lie himself, there has been close ties between Lie theory and differential equations. Thus it is not surprising that one finds that Lie theory and control are also closely connected. This „triangle“ is the subject of this set of notes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Flanders, Differential Forms, Academic Press, 1963, New York.
J. Wei and E. Norman,“On the Global Representation of the Solutions of Linear Differential Equations as a Product of Exponentials,” Proc. Am. Math. Soc., April 1964.
R. F. Gantmacher, Theory of Matrices, Chelsea, New York, 1959.
H. Samelson, Notes on Lie Algebras, Van Nostrand, New York, 1969.
M. Hausner and J. T. Schwartz, Lie Groups ; Lie Algebras, Gordon and Breach, London, 1968.
R. W. Brockett, “Lie Theory and Control Systems Defined on Spheres,” SIAM J. on Applied Math., Vol. 24, No. 5, Sept. 1973.
W. L. Chow, “Uber Systeme Von linearen partiellen Differentialgleichungen erster Ordinung,” Math. Ann. Vol. 117, (1939), pp. 98–105.
R. W. Brockett, “System Theory on Group Manifolds and Coset Spaces,” SIAM Journal on Control, Vol. 10, No. 2, May 1972, pp. 265–284.
S. Jonhansen (This Volume).
V. Jurdjevic and H. J. Sussmann, “Control Systems on Lie Groups,” J. of Differential Equations, Vol. 12, No. 2, (1972) pp. 313–329.
R. Hirschorn, Ph.D. Thesis, Applied Mathematics, Harvard University, 1973.
R. E. Rink and R. R. Mohler, “Completely Controllable Bilinear Systems,” SIAM J. on Control, Vol. 6, No. 3, 1968.
P. d’A lessandro, A. Isidori and A. Ruberti, “Realization and Structure Theory of Bilinear Dynamical Systems,” (to appear, SIAM J. on Control).
R. W. Brockett, “On the Algebraic Structure of Bilinear Systems,” in Variable Structure Control Systems, (R. Mohler and A. Ruberti, eds.) Academic Press, 1972, pp. 153–168.
A. J. Krener, A Generalization of the Pontryagin Maximal Principle and the Bang-Bang Principle, Ph.D. Thesis, Dept. of Mathematics, University of California, Berkeley, 1971.
H. J. Sussmann, “The Bang-Bang Problem for Linear Control Systems in Gl(n),” SIAM J. on Control, Vol. 10, No. 3, Aug. 1972
R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, New York, 1970.
R. Debye, “Polar Molecules,” The Chemical Catalogue Co., New York, 1929.
C. P. Poole and H. A. Farach, Relaxation in Magnetic Resonance, Academic Press, 1971.
F. Perrin, “Etude Mathematique du Mouvement Brownien de Rotation,” Ann. Ecole Norm. Sup., Vol. 45, (1928), pp. 1–51.
R. F. Fox, Physical Applications of Multiplicative Stochastic Processes, J. of Mathematical Physics, Vol. 14, No. 1, Jan. 1973, pp. 20–25.
V. A. Tutubalin, “Multimode Waveguides and Probability Distributions on a Symplectic Group,” Theory of Probability and Its Applications, Vol. 16, No. 4, pp. 631–642, 1971.
G. F. Carrier, “Stochastically Driven Dynamical Systems,” J. of Fluid Mechanics, Vol. 44, Part 2 (1970) pp. 249–264.
Martin Clark, (This Volume).
R. W. Brockett and J. C. Willems, “Average Value Criteria for Stochastic Stability,” Stability of Stochastic Dynamical Systems, (ed. Ruth Curtain), Springer-Verlag, 1972.
H. P. McKean, Stochastic Integrals, Academic Press, New York, 1969.
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1960.
M. E. Rose, Elementary Theory of Angular Momentum, J. Wiley, New York, 1957.
L. Hormander, “Hypoelliptic Second Order Differential Equations,” Acta. Math., 119: 147–171, 1967.
M. G. Krein, “Introduction to the Geometry of Indefinite J-Spaces and to the Theory of Operators in Those Spaces,” Am. Math. Soc. Translations, (Series 2) Vol. 93, 1970, pp. 103–176.
I. M. Gel’fand and V. B. Lidskii, “On the Structure of the Regions of Stability of Linear Canonical Systems of Differential Equations with Periodic Coefficients,” Am. Math. Soc. Transl. Series 2, Vol. 8, (1958) pp. 143–182.
R. W. Brockett and A. Rahimi, “Lie Algebras and Linear Differential Equations,” in Ordinary Differential Equations, (L. Weiss ed.) Academic Press, New York, 1972, pp. 379.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1973 D. Reidel Publishing Company, Dordrecht
About this paper
Cite this paper
Brockett, R.W. (1973). Lie Algebras and Lie Groups in Control Theory. In: Mayne, D.Q., Brockett, R.W. (eds) Geometric Methods in System Theory. NATO Advanced Study Institutes Series, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2675-8_2
Download citation
DOI: https://doi.org/10.1007/978-94-010-2675-8_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-2677-2
Online ISBN: 978-94-010-2675-8
eBook Packages: Springer Book Archive