Linear-Quadratic Problems and the Riccati Equation

  • A. H. W. Geerts
  • M. L. J. Hautus
Part of the Progress in Systems and Control Theory book series (PSCT, volume 2)


Linear-Quadratic (LQ) control problems have been investigated intensively since the fundamental and seminal paper of R.E. Kalman in 1960 ([KA]). In that paper, it was shown that the Riccati Equation plays an important role for the LQ-Problem. It is the purpose of the present paper to give an overview of the results relating the Riccati Equation with the LQ-Problem. LQ problems are also treated using the Hamiltonian matrix instead of the Riccati Equation. This will not be discussed in this paper. Neither will we deal with the use of Riccati equations outside of the LQ-problem context, e.g. in differential games and the H-optimization problem.


Riccati Equation Imaginary Axis Singular Case Dissipation Inequality Static State Feedback 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [G&H]
    A.H.W. Geerts & M.L.J. Hautus, “The output-stabilizable subspace and linear optimal control”, Proc. Internat. Symp. Math. Th. of Netw. & Syst., vol. 8, pp.1990.Google Scholar
  2. [G-1]
    Ton Geerts, “A necessary and sufficient condition for solvability of the linear-quadratic control problem without stability”, Syst. & Contr. Lett., vol. 11, pp. 47–51, 1988.CrossRefGoogle Scholar
  3. [G-2]
    Ton Geerts, “All optimal controls for the singular linear-quadratic problem without stability; a new interpretation of the optimal cost”, Lin. Al g. & Appl., vol. 116, pp. 135–181, 1989.CrossRefGoogle Scholar
  4. [G-3]
    Ton Geerts, “The computation of optimal controls for the singular linear-quadratic problem that yield internal stability”, EUT Report 89-WSK-03, Eindhoven University of Technology, 1989.Google Scholar
  5. [G-4]
    Ton Geerts, Structure of Linear-Quadratic Control, Ph.D. Thesis, Eindhoven, 1989.Google Scholar
  6. [H&S]
    M.L.J. Hautus & L.M. Silverman, “System structure and singular control”, Lin. Alg. & Appl., vol. 50, pp. 369–402, 1983.CrossRefGoogle Scholar
  7. [Ka]
    R.E. Kalman, “Contributions to the theory of optimal control”, Bol. Soc. Mat. Mex., vol. 5, pp. 102–199, 1960.Google Scholar
  8. [Mo]
    B.P. Molinari, “The time-invariant linear-quadratic optimal control problem”, Automatica, vol. 13, pp. 347–357, 1977.CrossRefGoogle Scholar
  9. [Po]
    V.M. Popov, “Hyperstability and optimality of automatic systems with several control functions”, Rev. Roumaine Sci. Tech. Ser. Electrotech. Energet., vol. 9, pp. 629–690, 1964.Google Scholar
  10. [Sc]
    J.M. Schumacher, “The role of the dissipation matrix in singular optimal control”, Syst. & Contr. Lett., vol. 2, pp. 262–266, 1983.CrossRefGoogle Scholar
  11. [Wi]
    J.C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation”, IEEE Trans. Automat. Contr., vol. AC-16, pp. 621–634, 1971.CrossRefGoogle Scholar
  12. [WKS]
    J.C. Willems, A. Kitapçi & L.M. Silverman, “Singular optimal control: A geometric approach”, SIAM J. Contr. & Opt., vol., 24, pp. 323–337, 1986.CrossRefGoogle Scholar
  13. [Wo]
    W.M. Wonham, Linear Multivariable Control: A Geometric Approach, Springer, New York, 1979.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • A. H. W. Geerts
    • 1
  • M. L. J. Hautus
    • 1
  1. 1.Dept. of Math. & Comp. Sci.Eindhoven University of technologyThe Netherlands

Personalised recommendations