Journal of Optimization Theory and Applications

, Volume 140, Issue 2, pp 301–314 | Cite as

Revisit of Linear-Quadratic Optimal Control

  • M. Pachter


The classical finite-dimensional linear-quadratic optimal control problem is revisited. A new linear-quadratic control problem with linear state penalty terms but without quadratic state penalty terms, is introduced. An optimal control exists and the closed-form optimal solution is given. It is remarkable that feedback action plays no role and state information does not feature in the optimal control. The optimal cost function, rather than being quadratic, is linear in the initial state.


Linear quadratic control Feedback control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kalman, R.E.: Contributions to the theory of optimal control. Mex. Math. Soc. Bull. 2(5), 102–119 (1960) MathSciNetGoogle Scholar
  2. 2.
    Athans, M. (ed.): IEEE Trans. Autom. Control. December 1971. Special Issue on the Linear-Quadratic-Gaussian Estimation and Control Problem Google Scholar
  3. 3.
    Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation and Control. Halsted Press, New York (1975) Google Scholar
  4. 4.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Dover, New York (1998) Google Scholar
  5. 5.
    Stengel, R.F.: Optimal Control and Estimation. Dover, New York (1994) Google Scholar
  6. 6.
    Lewis, F.L., Syrmos, V.L.: Optimal Control and Estimation. Prentice Hall, New York (1992) Google Scholar
  7. 7.
    Brogan, W.L.: Modern Control Theory. Prentice Hall, New York (1991) Google Scholar
  8. 8.
    Brockett, R.W.: Finite Dimensional Linear Systems. Wiley, New York (1970), pp. 136–141 Google Scholar
  9. 9.
    Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear-Quadratic Methods. Prentice Hall, New York (1998) Google Scholar
  10. 10.
    Jacobson, D.H.: Extensions of Linear-Quadratic Control, Optimization, and Matrix Theory. Mathematics in Science and Engineering, vol. 133. Academic Press, San Diego (1977) CrossRefGoogle Scholar
  11. 11.
    Jacobson, D.H., Martin, D., Pachter, M.: Extensions of Linear-Quadratic Control. Springer, Berlin (1980) Google Scholar
  12. 12.
    Bellman, R.E.: Linear Equations and Quadratic Criteria. Introduction to the Mathematical Theory of Control Processes, vol. 1. Academic Press, San Diego (1967) Google Scholar
  13. 13.
    Clements, D.J., Anderson, B.D.O.: Singular Optimal Control: the Linear-Quadratic Problem. Springer, New York (1978) Google Scholar
  14. 14.
    Engwerda, J.: LQ Dynamic Optimization and Differential Games. Wiley, New York (2006) Google Scholar
  15. 15.
    Dorato, P., Abdallah, C.T., Cerone, V.: Linear-Quadratic Control: an Introduction. Macmillan, New York (1994) Google Scholar
  16. 16.
    Daiuto, B.J., Hartley, T.T.: The Hyperbolic Map and Applications to the Linear Quadratic Regulator. Lecture Notes in Control and Information Sciences, vol. 110. Springer, Berlin (1989) Google Scholar
  17. 17.
    Mehrmann, V.L.: The Autonomous Linear Quadratic Control Problem. Lecture Notes in Control and Information Sciences, vol. 163. Springer, Berlin (1991). ISBN:3540541705 Google Scholar
  18. 18.
    Sima, V.: Algorithms for Linear-Quadratic Optimization. Marcel Dekker, New York (1996) Google Scholar
  19. 19.
    Abou-Kandil, H., Freiling, G., Ionescu, V.: Matrix Riccati Equations in Control and Systems Theory. Birkhauser, Basel (2003) Google Scholar
  20. 20.
    Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, London (1995) Google Scholar
  21. 21.
    Ionescu, V., Oara, C., Weiss, M.: Generalized Riccati Theory and Robust Control. Wiley, New York (1998) Google Scholar
  22. 22.
    Bittani, S., Laub, A.J., Willems, J.C.: The Riccati Equation. Springer, Berlin (1991) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringAir Force Institute of Technology, AFIT/ENGWright-Patterson Air Force BaseUSA

Personalised recommendations