zbMATH — the first resource for mathematics

Discrete-time indefinite stochastic linear quadratic optimal control with second moment constraints. (English) Zbl 1407.93435
Summary: This paper studies the discrete-time stochastic linear quadratic (LQ) problem with a second moment constraint on the terminal state, where the weighting matrices in the cost functional are allowed to be indefinite. By means of the matrix Lagrange theorem, a new class of generalized difference Riccati equations (GDREs) is introduced. It is shown that the well-posedness, and the attainability of the LQ problem and the solvability of the GDREs are equivalent to each other.

93E20 Optimal stochastic control
93C55 Discrete-time control/observation systems
Full Text: DOI
[1] Kalman, R. E., Contributions to the theory of optimal control, Boletín de la Sociedad Matemática Mexicana, 5, 2, 102-119, (1960)
[2] Wonham, W. M., On a matrix Riccati equation of stochastic control, SIAM Journal on Control and Optimization, 6, 4, 681-697, (1968) · Zbl 0182.20803
[3] de Souza, C. E.; Fragoso, M. D., On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control, Systems and Control Letters, 14, 3, 233-239, (1990) · Zbl 0701.93106
[4] Chen, B. S.; Zhang, W., Stochastic \(H_2 / H_\infty\) control with state-dependent noise, IEEE Transactions on Automatic Control, 49, 1, 45-57, (2004) · Zbl 1365.93539
[5] Zhang, W.; Chen, B. S., On stabilizability and exact observability of stochastic systems with their applications, Automatica, 40, 1, 87-94, (2004) · Zbl 1043.93009
[6] Anderson, B. D. O.; Moore, J. B., Optimal Control Linear Quadratic Mathods, (1989), New York, NY, USA: Prentice-Hall, New York, NY, USA
[7] Lewis, F. L., Optimal Control, (1986), New York, NY, USA: John Wiley and Sons, New York, NY, USA
[8] Huang, Y.; Zhang, W.; Zhang, H., Infinite horizon linear quadratic optimal control for discrete-time stochastic systems, Asian Journal of Control, 10, 5, 608-615, (2008)
[9] Chen, S.; Li, X. J.; Zhou, X. Y., Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36, 5, 1685-1702, (1998) · Zbl 0916.93084
[10] Rami, M. A.; Zhou, X. Y., Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls, IEEE Transactions on Automatic Control, 45, 6, 1131-1143, (2000) · Zbl 0981.93080
[11] Yao, D. D.; Zhang, S.; Zhou, X. Y., Stochastic linear-quadratic control via semidefinite programming, SIAM Journal on Control and Optimization, 40, 3, 801-823, (2001) · Zbl 1055.93077
[12] Ait Rami, M.; Moore, J. B.; Zhou, X. Y., Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM Journal on Control and Optimization, 40, 4, 1296-1311, (2001) · Zbl 1009.93082
[13] Ferrante, A.; Marro, G.; Ntogramatzidis, L., A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions, Automatica, 41, 8, 1359-1366, (2005) · Zbl 1086.93020
[14] Bertsimas, D.; Brown, D. B., Constrained stochastic LQC: a tractable approach, IEEE Transactions on Automatic Control, 52, 10, 1826-1841, (2007) · Zbl 1366.93699
[15] Hu, Y.; Zhou, X. Y., Constrained stochastic LQ control with random coefficients, and application to portfolio selection, SIAM Journal on Control and Optimization, 44, 2, 444-466, (2005) · Zbl 1210.93082
[16] Ko, S.; Bitmead, R. R., Optimal control for linear systems with state equality constraints, Automatica, 43, 9, 1573-1582, (2007) · Zbl 1128.93365
[17] Kojima, A.; Morari, M., LQ control for constrained continuous-time systems, Automatica, 40, 7, 1143-1155, (2004) · Zbl 1051.93035
[18] Zhou, Z.; Randy, C., An algorithm for state constrained stochastic linear-quadratic control, Proceedings of the American Control Conference
[19] Huang, Y. L.; Zhang, W. H., Study on stochastic linear quadratic optimal control with constraint, Acta Automatica Sinica, 32, 2, 246-254, (2006)
[20] Li, G.; Zhang, W., Discrete-time indefinite stochastic linear quadratic optimal control with equality constraints, Proceeding of the Chinese Control and Decision Conference
[21] Dong, H.; Wang, Z.; Ho, D. W. C.; Gao, H., Variance-constrained \(H_\infty\) filtering for a class of nonlinear time-varying systems with multiple missing measurements: the finite-horizon case, IEEE Transactions on Signal Processing, 58, 5, 2534-2543, (2010) · Zbl 1391.93233
[22] Ma, L.; Bo, Y.; Zhou, Y.; Guo, Z., Error variance-constrained \(H_\infty\) filtering for a class of nonlinear stochastic systems with degraded measurements: the finite horizon case, International Journal of Systems Science, 43, 12, 2361-2372, (2012) · Zbl 1305.93189
[23] Ait Rami, M.; Chen, X.; Zhou, X. Y., Discrete-time indefinite LQ control with state and control dependent noises, Journal of Global Optimization, 23, 3-4, 245-265, (2002) · Zbl 1035.49024
[24] Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, (1996), Belmont, Mass, USA: Athena Scientific, Belmont, Mass, USA
[25] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM Journal on Applied Mathematics, 17, 2, 434-440, (1969) · Zbl 0265.15002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.