Switching surfaces and groebner bases

  • Tryphon Georgiou
  • Allen Tannenbaum
Part A Learning And Computational Issues
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 241)


A number of problems in control can be reduced to finding suitable real solutions of algebraic equations. In particular, such a problem arises in the context of switching surfaces in optimal control. Recently, a powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested, namely the use of Groebner bases. In this note, we apply the Groebner basis technique to find effective solutions to the classical problem of time-optimal control.


Algebraic Geometry Polynomial Equation Switching Strategy Monomial Ideal Symbolic Manipulation 
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  1. [1]
    M. Athans and P. Falb, Optimal Control: An Introduction to the Theory and its Applications, New York: McGraw-Hill, 1966.Google Scholar
  2. [2]
    D. Bayer and D. Mumford, “What can be computed in algebraic geometry?,” in Computational Algebraic Geometry and Commutative Algebra, edited by D. Eisenbud and L. Robbiano, Cambridge University Press, Cambridige, 1993, pages 1–48.Google Scholar
  3. [3]
    T. Becker and V. Weispfenning, Groebner Bases, Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  4. [4]
    D. Cox, J. LIttle, and D. O'shea, Ideals, Varieties, and Algorithms, Second Edition Springer-Verlag, New York, 1997.Google Scholar
  5. [5]
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1976.Google Scholar
  6. [6]
    Y. Kannai, “Causality and stability of linear systems described by partial differential operators,” SIAM J. Control Opt. 20: 669–674, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    E.B. Lee and L. Markus, Foundations of Optimal Control Theory, Malabar, FL:Krieger, 1967.zbMATHGoogle Scholar
  8. [8]
    L.Y. Pao and G.F. Franklin: “Proximate time-optimal control of thirdorder servomechanisms,” IEEE Transactions on Automatic Control, 38(4): 560–280, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Tannenbaum, Invariance and Systems Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics 845, Springer-Verlag, New York, 1981.Google Scholar
  10. [10]
    A. Tannenbaum and Y. Yomdin, “Robotic manipulators and the geometry of real semi-algebraic sets,” IEEE Journal of Robotics and Automation RA-3 (1987), 301–308.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Tryphon Georgiou
  • Allen Tannenbaum
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolis

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