Abstract
Some geometrical aspects of Hamiltonian systems arising in optimal control problems are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature Cited
R. W. Brockett, “Lie theory and control systems defined on spheres,” SIAM J. Appl. Math.,25, No. 2, 213–225 (1973).
J. Baillieul, “Geometric methods for nonlinear control problem,” J. Optim. Theory Appl.,25, No. 4, 519–548 (1978).
R. W. Brockett, “System theory on group manifolds and coset spaces,” SIAM J. Contr. Optim.,10, No. 2, 265–284 (1972).
L. E. Faibusovich, “On linear dynamic systems with symmetries,” Avtomat. Telemekh., No. 5, 82–89 (1983).
L. E. Faibusovich, “On stabilization of infinite-dimensional linear dynamic systems by the Kalman-Letov method,” Avtomat. Telemekh., No. 2, 52–58 (1985).
L. E. Faibusovich, “Algebraic Riccati equation and symplectic algebra,” Int. J. Contr.,43, No. 3, 781–792 (1986).
L. E. Faibusovich, “Existence and uniqueness of extremal solutions of the Riccati equation and symplectic, geometry,” Funkts. Anal. Ego Prilozhen.,19, No. 1, 85–86 (1985).
L. E. Faibusovich, “On the symplectic structure of the Riccati operator equation,” Kibernet. Vychisl. Tekh., No. 65, 62–67 (1985).
L. E. Faibusovich, “Riccati equation, multidimensional Euler equation, and isospectral deformation,” Kibernet. Vychisl. Tekh., No. 69, 37–40 (1986).
L. E. Faibusovich, “Analytical design of a linear regulator and multidimensional analog of the equations of rigid body dynamics,” Avtomat. Telemekh., No. 7, 81–90 (1987).
G. N. Yakovenko, “Trajectory synthesis of optimal control,” Avtomat. Telemekh., No. 6, 5–12 (1972).
M. Hazewinkel, “Control and filtering of a class of nonlinear but ‘homogeneous’ system,” Lect. Notes Inform. Sci. Contr.,39, 123–146 (1982).
Yu. N. Andreev, “Differential-geometric methods in control theory,” Avtomat. Telemekh., No. 9, 5–46 (1982).
V. Guillemin and S. Sternberg, “The moment map and collective motion,” Ann. Phys.,127, 52–65 (1980).
V. A. Yatsenko, “Dynamic equivalent systems in the solution of some optimal control problems,” Avtomatika, No. 4, 59–65 (1984).
F. Griffiths, External Differential Systems and Variational Calculus [Russian translation], Mir, Moscow (1986).
A. M. Vershik and V. Ya. Gershkovich, “Geodesic flow on SL(2, R) with noholonomic constraints,” Zap. Nauchn. Seminarov LOMI,155, 7–17 (1985).
A. I. Kukhtenko, “On physics and cybernetics,” Kibernetika, No. 4, 133–138 (1981).
B. A. Dubrovin, S. P. Novikov, and A. P. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1986).
A. Weinstein, “The local structure of Poisson manifolds,” J. Diff. Geom.,18, No. 3, 525–577 (1983).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).
H. J. Sussman, “Lie brackets, real analyticity and geometric control theory,” Differential Geometric Control Theory, Birkhauser, Boston (1983), pp. 1–116.
A. G. Butkovskii and Yu. I. Samoilenko, Control of Quantum-Mechanical Processes [in Russian], Nauka, Moscow (1984).
Additional information
Translated from Kibernetika, No. 2, pp. 73–77, March–April, 1989.
Rights and permissions
About this article
Cite this article
Faibusovich, L.E. Collective Hamiltonian method in optimal control problems. Cybern Syst Anal 25, 230–237 (1989). https://doi.org/10.1007/BF01070131
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01070131