On a class of rational matrix differential equations arising in stochastic control

Linear Algebra and Its Applications

Volumn 379, Issue 1-3 SPEC. ISS, 2004, Pages 43-68

  Freiling, G ^a   Hochhaus, A ^a  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

Author keywords

Comparison theorem; Existence and convergence results; Generalized Riccati differential equations; Generalized stabilizability and detectability; Rational matrix differential equations

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FUNCTION EVALUATION; LINEAR EQUATIONS; OPTIMAL CONTROL SYSTEMS; PROBABILITY; RICCATI EQUATIONS; STABILITY; STOCHASTIC CONTROL SYSTEMS; THEOREM PROVING;

GENERALIZED RICCATI DIFFERENTIAL EQUATIONS; GENERALIZED STABIZABILITY AND DETECTABILITY; MONOTONICITY; RATIONAL MATRIX DIFFERENTIAL EQUATIONS;

MATRIX ALGEBRA;

EID: 0742284369     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(02)00651-1     Document Type: Conference Paper

References (36)

1. Abou-Kandil H., Freiling G., Jank G. Solution and asymptotic behavior of coupled Riccati equations in jump linear systems. IEEE Trans. Automat. Control. 39:1994;1631-1636.
2. Ahlbrandt C.D., Peterson A.C. Discrete Hamiltonian systems. 1996;Kluwer Academic Publishers Group, Dordrecht.
3. Ait Rami M., Chen X., Moore J.B., Zhou X.Y. Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls. IEEE Trans. Automat. Control. 46:2001;428-440.
4. Ait Rami M., Moore J.B., Zhou X.Y. Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. Control Optim. 40:2001/2002;1296-1311.
5. Ait Rami M., Zhou X.Y. Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Trans. Automat. Control. 45:2000;1131-1143.
6. Albert A. Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17:1969;434-440.
7. T. Ando, Matrix quadratic equations, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo 1988.
8. Arendt W., Batty C.J.K., Hieber M., Neubrander F. Vector-valued Laplace Transforms and Cauchy Problems. 2001;Birkhäuser Verlag, Basel.
9. Chen S., Li X., Zhou X.Y. Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim. 36:1998;1685-1702.
10. Chen S., Yong J. Stochastic linear quadratic optimal control problems. Appl. Math. Optim. 43:2001;21-45.
11. Chen S., Zhou X.Y. Stochastic linear quadratic regulators with indefinite control weight costs II. SIAM J. Control Optim. 39:2000;1065-1081.
12. Clements D.J., Wimmer H.K. Monotonicity of the optimal cost in the discrete-time regulator problem and Schur complements. Automatica J. IFAC. 37:2001;1779-1786.
13. Damm T., Hinrichsen D. Newton's method for a rational matrix equation occuring in stochastic control. Linear Algebra Appl. 332/334:2001;81-109.
14. T. Damm, D. Hinrichsen, Newton's method for concave operators with resolvent positive derivatives in ordered Banach spaces, Linear Algebra Appl., in press.
15. Dragan V., Morozan T. Global solutions to a game-theoretic Riccati equation of stochastic control. J. Differential Equations. 138:1997;328-350.
16. Dragan V., Morozan T. Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise. Stochastic Anal. Appl. 20:2002;33-92.
17. V. Dragan, T. Morozan, Systems of matrix rational differential equations arising in connection with linear stochastic systems with markovian jumping, Preprint No. 9/2000, Institutul de Matematica al academiei Romane, 2000.
18. Feng X., Loparo K.A., Ji Y., Chizeck H.J. Stochastic stability properties of jump linear systems. IEEE Trans. Automat. Control. 37:1992;38-53.
19. Fragoso M.D., Costa O.L.V., de Souza C.E. A new approach to linearly perturbed Riccati equations arising in stochastic control. Appl. Math. Optim. 37:1998;99-126.
20. G. Freiling, A. Hochhaus, Basic properties of a class of rational matrix differential equations, in: Proceedings of the European Control Conference (ECC 2001), Porto, 4-7 September 2001.
21. G. Freiling, A. Hochhaus, Properties of the solutions of rational matrix difference equations. Advances in difference equations, IV, Comput. Math. Appl., in press.
22. Freiling G., Jank G. Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations. J. Dynam. Control Systems. 2:1996;529-547.
23. Freiling G., Jank G., Abou-Kandil H. Generalized Riccati difference and differential equations. Linear Algebra Appl. 241/243:1996;291-303.
24. Harville D.A. Matrix Algebra From a Statistician's Perspective. 1997;Springer-Verlag, New York.
25. ∞ SIAM J. Control Optim. 36:1998;1504-1538.
26. Kolmanovskii V.B., Richard J.-P. Some Riccati equations in the stability study of dynamical systems with delays. Neural Parallel Sci. Comput. 7:1999;235-252.
27. Kučera V. Algebraic Riccati equation: Hermitian and definite solutions. The Riccati equation. in: Comm. Control Engrg. Ser. 1991;53-88 Springer, Berlin.
28. Lancaster P., Rodman L. Algebraic Riccati equations. 1995;Clarendon Press, Oxford.
29. Lancaster P., Tismenetsky M. The Theory of Matrices. 1985;Academic Press Inc. Orlando.
30. Ran A.C.M., Vreugdenhil R. Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems. Linear Algebra Appl. 99:1988;63-83.
31. Reid W.T. Riccati Differential Equations. 1972;Academic Press, New York, London.
32. E.I. Verriest, Robust stability and stabilization: from linear to nonlinear, Proc. IFAC Workshop, Ancona, 2000, pp. 184-195.
33. 0-semigroups on AL-spaces Semigroup Forum. 38:1989;263-266.
34. Wonham W.M. On a matrix Riccati equation of stochastic control. SIAM J. Control. 6:1968;681-697.
35. Yong J., Zhou X.Y. Stochastic Controls. 1999;Springer-Verlag, New York.
36. Zeidler E. Nonlinear Functional Analysis and its Applications I. Fixed-point Theorems. 1986;Springer- Verlag, New York, Berlin.