Abstract
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.
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References
Kalman, R.E.: Contributions to the theory of optimal control. Mex. Math. Soc. Bull. 2(5), 102–119 (1960)
Athans, M. (ed.): IEEE Trans. Autom. Control. December 1971. Special Issue on the Linear-Quadratic-Gaussian Estimation and Control Problem
Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation and Control. Halsted Press, New York (1975)
Kirk, D.E.: Optimal Control Theory: An Introduction. Dover, New York (1998)
Stengel, R.F.: Optimal Control and Estimation. Dover, New York (1994)
Lewis, F.L., Syrmos, V.L.: Optimal Control and Estimation. Prentice Hall, New York (1992)
Brogan, W.L.: Modern Control Theory. Prentice Hall, New York (1991)
Brockett, R.W.: Finite Dimensional Linear Systems. Wiley, New York (1970), pp. 136–141
Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear-Quadratic Methods. Prentice Hall, New York (1998)
Jacobson, D.H.: Extensions of Linear-Quadratic Control, Optimization, and Matrix Theory. Mathematics in Science and Engineering, vol. 133. Academic Press, San Diego (1977)
Jacobson, D.H., Martin, D., Pachter, M.: Extensions of Linear-Quadratic Control. Springer, Berlin (1980)
Bellman, R.E.: Linear Equations and Quadratic Criteria. Introduction to the Mathematical Theory of Control Processes, vol. 1. Academic Press, San Diego (1967)
Clements, D.J., Anderson, B.D.O.: Singular Optimal Control: the Linear-Quadratic Problem. Springer, New York (1978)
Engwerda, J.: LQ Dynamic Optimization and Differential Games. Wiley, New York (2006)
Dorato, P., Abdallah, C.T., Cerone, V.: Linear-Quadratic Control: an Introduction. Macmillan, New York (1994)
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)
Mehrmann, V.L.: The Autonomous Linear Quadratic Control Problem. Lecture Notes in Control and Information Sciences, vol. 163. Springer, Berlin (1991). ISBN:3540541705
Sima, V.: Algorithms for Linear-Quadratic Optimization. Marcel Dekker, New York (1996)
Abou-Kandil, H., Freiling, G., Ionescu, V.: Matrix Riccati Equations in Control and Systems Theory. Birkhauser, Basel (2003)
Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, London (1995)
Ionescu, V., Oara, C., Weiss, M.: Generalized Riccati Theory and Robust Control. Wiley, New York (1998)
Bittani, S., Laub, A.J., Willems, J.C.: The Riccati Equation. Springer, Berlin (1991)
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Communicated by A. Miele.
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Pachter, M. Revisit of Linear-Quadratic Optimal Control. J Optim Theory Appl 140, 301–314 (2009). https://doi.org/10.1007/s10957-008-9449-4
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DOI: https://doi.org/10.1007/s10957-008-9449-4