Geodesic
On the theory of the matrix Riccati equation
Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 341-354 
M. I. Zelikin. On the theory of the matrix Riccati equation. Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 341-354. http://geodesic.mathdoc.fr/item/SM_1992_73_2_a2/
@article{SM_1992_73_2_a2,
     author = {M. I. Zelikin},
     title = {On the theory of the matrix {Riccati} equation},
     journal = {Sbornik. Mathematics},
     pages = {341--354},
     year = {1992},
     volume = {73},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_73_2_a2/}
}
TY  - JOUR
AU  - M. I. Zelikin
TI  - On the theory of the matrix Riccati equation
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 341
EP  - 354
VL  - 73
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1992_73_2_a2/
LA  - en
ID  - SM_1992_73_2_a2
ER  - 
%0 Journal Article
%A M. I. Zelikin
%T On the theory of the matrix Riccati equation
%J Sbornik. Mathematics
%D 1992
%P 341-354
%V 73
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1992_73_2_a2/
%G en
%F SM_1992_73_2_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

Various approaches to the study of the matrix Riccati equation with variable coefficients are investigated. It is proved that the complexification of such an equation determines a flow on the Seigel generalized upper half-plane and on each of the strata forming its boundary. The concept of the matrix cross-ratio of a quadruple of points of the Grassmann manifold $G_n(\mathbf R^{2n})$ is introduced, and applications of it are given. In particular, a criterion is given for the preservation of the isoclinic property for a pair of planes that are displaced by the flow on $G_n(\mathbf R^{2n})$ determined by a matrix Riccati equation with variable coefficients. Bilinear optimal control problems with a quadratic quality criterion are considered. The corresponding extremals are found along with the matrix Riccati equations determined by them.

[1] Zelikin M. I., Optimalnoe upravlenie i variatsionnoe ischislenie, Izd-vo MGU, M., 1985 | MR | Zbl

[2] Herman R., Martin C., “Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations, I”, Math. Systems Theory, 1982, no. 15, 277–284 | MR | Zbl

[3] Herman R., Martin C., “Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations, II”, Math. Systems Theory, 1983, no. 16, 297–308 | DOI | MR

[4] Shayman M., “Phase portrait of the matrix Riccati equation”, SIAM J. Control Opt., 24 (1986), 1–65 | DOI | MR | Zbl

[5] Arnold V. I., “Teoremy Shturma v simplekticheskoi geometrii”, Funktsion. analiz. i ego pril., 19:4 (1985), 1–10 | MR

[6] Andrianov A. N., Zhuravlev V. G., Modulyarnye formy i operatory Gekke, Nauka, M., 1990 | MR

[7] Wong Y. C., “Isoclinic $n$-planes in Euclidean $2n$-space, Clifford parallels in elliptic $(2n-1)$-space and the Hurwitz matrix equations”, Mem. Amer. Math. Soc., 41 (1961), 1–112 | MR

[8] Wolf J., “Geodesic spheres in Grassmann manifolds”, Illinois Journal of Math., 7:3 (1963), 425–462 | MR

[9] Brockett R. W., “Lie algebras and Lie groups in control theory”, Geometric Methods in Control Theory, eds. D. Q. Mayne, R. W. Brockett, Reidel, Dordrecht, The Netherlands, 1973 | Zbl

[10] Brockett R. W., Millman R. S., Sussmann H. J., Differential geometric control theory, Birkhauser, Boston, 1983 | MR | Zbl

[11] Vershik A. M., Gershkovich V. Ya., “Negolonomnye dinamicheskie sistemy, geometriya raspredelenii i variatsionnye zadachi”, Itogi nauki i tekhn., 16, VINITI, M., 1987, 5–85 | MR

[12] Moser J., “Three integrable Hamiltonian systems, connected with isospectral deformations”, Adv. Math., 16:2 (1975), 197–220 | DOI | MR | Zbl