Abstract
In this paper, we consider discrete-time systems. We study conditions under which there is a unique control that minimizes a general quadratic cost functional. The system considered is described by a linear time-invariant recurrence equation in which the number of inputs equals the number of states. The cost functional differs from the usual one considered in optimal control theory, in the sense that we do not assume that the weight matrices considered are semipositive definite. For both a finite planning horizon and an infinite horizon, necessary and sufficient solvability conditions are given. Furthermore, necessary and sufficient conditions are derived for the existence of a solution for an arbitrary finite planning horizon.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference
Pindyck, R. A.,Optimal Planning for Economic Stabilization, North Holland, Amsterdam, Netherlands, 1973.
Chow, G. C.,Analysis and Control of Dynamic Economic Systems, John Wiley and Sons, New York, New York, 1975.
Pitchford, J. D., andTurnovsky, S. J.,Applications of Control Theory to Economic Analysis, North Holland, Amsterdam, Netherlands, 1977.
Preston, A. J., andPagan, A. R.,The Theory of Economic Policy, Cambridge University Press, New York, New York, 1982.
De Zeeuw, A. J.,Difference Games and Linked Econometric Policy Models, PhD Thesis, Tilburg University, Tilburg, Netherlands, 1984.
Engwerda, J. C.,The Solution of the Infinite-Horizon Tracking Problem for Discrete-Time Systems Possessing an Exogenous Component, Journal of Economic Dynamics and Control, Vol. 14, pp. 741–762, 1990.
Engwerda, J. C.,The Indefinite LQ-Problem: Existence of a Unique Solution, Proceedings of the DGOR/OGOR Annual Conference, Aachen, Germany, Edited by K. W. Hansmann, A. Bachem, M. Jarke, W. E. Katzenberger, and A. Marusev, Springer Verlag, Berlin, Germany, pp. 217–223, 1993.
Başar, T., andOlsder, G. J.,Dynamic Noncooperative Game Theory, Academic Press, New York, New York, 1982.
Stoorvogel, A. A.,The H ∞-Control Problem: A State Space Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1992.
Levy, B. C.,Regular and Reciprocal Multivariate Stationary Gaussian Processes over Z Are Necessarily Markov, Journal of Mathematical Systems, Estimation, and Control., Vol. 2, pp. 133–154, 1992.
Engwerda, J. C., Ran, A. C. M., andRijkeboer, A. L.,Necessary and Sufficient Conditions for the Existence of a Positive-Definite Solution of the Matrix Equation X+A * X −1 A=Q, Linear Algebra and Its Applications, Vol. 186, pp. 255–277, 1993.
Engwerda, J. C.,The Square Indefinite LQ-Problem: Existence of a Unique Solution, Proceedings of the ECC-93 Conference, Groningen, Netherlands, Edited by J. W. Nieuwenhius, C. Praagman, and H. L. Trentelman, Vol. 1, pp. 329–333, 1993.
Willems, J. C.,Least Squares Stationary Optimal Control and the Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. 6, pp. 621–634, 1971.
Molinari, B. P.,The Time-Invariant Linear-Quadratic Optimal Control Problem, Automatica, Vol. 13, pp. 347–357, 1977.
Trentelman, H. L.,The Regular Free-Endpoint Linear Quadratic Problem with Indefinite Cost, SIAM Journal on Control and Optimization, Vol. 27, pp. 27–42, 1989.
Soethoudt, J. M., andTrentelman, H. L.,The Regular Indefinite Linear Quadratic Problem with Linear Endpoint Constraints, Systems and Control Letters, Vol. 12, pp. 23–31, 1989.
Jonckheere, E. A., andSilverman, L. M.,Spectral Theory of the Linear-Quadratic Optimal Control Problem: Discrete-Time Single-Input Case, IEEE Transactions on Circuits and Systems, Vol. 25, pp. 810–825, 1978.
Jonckheere, E. A., andSilverman, L. M.,Spectral Theory of the Linear-Quadratic Optimal Control Problem: A New Algorithm for Spectral Computations, IEEE Transactions on Automatic Control, Vol. 25, pp. 880–888, 1980.
Lancaster, P., Ran, A. C. M., andRodman, L.,Hermitian Solutions of the Discrete Algebraic Riccati Equation, International Journal of Control, Vol. 44, pp. 777–802, 1986.
Ran, A. C. M., andRodman, L.,Stable Hermitian Solutions of Discrete Algebraic Riccati Equations, Mathematics of Control, Signals, and Systems, Vol. 5, pp. 165–193, 1992.
Ran, A. C. M., andTrentelman, H. L.,Linear-Quadratic Problems with Indefinite Cost for Discrete-Time Systems, SIAM Journal on Matrix Analysis and Applications, Vol. 14, pp. 776–797, 1993.
Pappas, T., Laub, A. J., andSandell, N. R.,On the Numerical Solution of the Discrete-Time Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. 25, pp. 631–641, 1980.
Molinari, B. P.,Nonnegativity of a Quadratic Functional, SIAM Journal on Control, Vol. 13, pp. 792–806, 1975.
Engwerda, J. C.,On the Existence of a Positive-Definite Solution of the Matrix Equation X+A T X −1 A=I, Linear Algebra and Its Applications, Vol. 194, pp. 91–109, 1993.
Rozanov, Y. A.,Stationary Random Processes, Holden-Day, San Francisco, California, 1967.
Engwerda, J. C.,The Indefinite LQ-Problem: The Finite Planning Horizon Case, Research Memorandum FEW 535, Department of Economics, Tilburg University, Tilburg, Netherlands, 1992.
Rappaport, D., andSilverman, L. M.,Structure and Stability of Discrete-Time Optimal System, IEEE Transactions on Automatic Control, Vol. 16, pp. 227–232, 1971.
Kailath, T.,Linear Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1980.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
The author dedicates this paper to the memory of his late grandfather Jacob Oosterwold.
Rights and permissions
About this article
Cite this article
Engwerda, J.C. Square indefinite LQ-problem: Existence of a unique solution. J Optim Theory Appl 90, 627–648 (1996). https://doi.org/10.1007/BF02189799
Issue Date:
DOI: https://doi.org/10.1007/BF02189799