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Matrix Riccati Equations in Optimal Control
School of Computer Sciences and Mathematics, Marist College, Poughkeepsie, USA
- 1 School of Computer Sciences and Mathematics, Marist College, Poughkeepsie, USA
Applied Mathematics·Volume 15 (2024)·Pages 199–213·Published 29 March 2024·DOI10.4236/am.2024.153011
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Abstract
In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.
KeywordsOptimal ControlMatrix Riccati EquationChange of Variable
- https://mathshistory.st-andrews.ac.uk/Biographies/Riccati
- Reid, W.T. (1980) Riccati Differential Equations. Academic Press, New York.
- Leitman, G. (1981) The Calculus of Variations and Optimal control. Plenum Press, NY & London. https://doi.org/10.1007/978-1-4899-0333-4
- Fraga, S., García de la Vega, J.M. and Fraga, E.S. (1999) The Schrodinger and Riccati Equations. In: Lecture Notes in Chemistry, Vol. 70, Springer-Verlag, Berlin.
- Scwabl, F. (1992) Quantum Mechanics. Springer, Berlin.
- Abou-Kandil, H., Freiling, G., Ionescu, V. and Jank, G. (2003) Matrix Riccati Equations in Control and Systems Theory. In: Systems & Control: Foundations & Applications, Birkhäuser, Basel, 299-410. https://doi.org/10.1007/978-3-0348-8081-7
- Kalman, R.E and Bucy, R.S. (1961) New Results in Linear Filtering and Prediction Theory. Journal of Fluids Engineering, 83, 95-108. https://doi.org/10.1115/1.3658902
- Boyle, P.P., Tian, W. and Guan, F. (2002) The Riccati Equation in Mathematical Finance. Journal of Symbolic Computation, 33, 343-355. https://doi.org/10.1006/jsco.2001.0508
- Ndiaye, M. (2022) The Riccati Equation, Differential Transform, Rational Solutions and Applications. Applied Mathematics, 13, 774-792. https://doi.org/10.4236/am.2022.139049
- Ince, E.L. (1956) Ordinary Differential Equations. Dover Publications, New York.
- Mack, J. and Strauss, A. (1981) Introduction to Optimal Control Theory. Springer Verlag, New York.
- Kirk, D.E. (2004) Optimal Control Theory: An Introduction. Dover Publications, New York.
- Kucera, V. (1973) A Review of the Matrix Riccati Equation. Kybernetica, 9, 42-61.