- Previous Article
- MCRF Home
- This Issue
-
Next Article
Asymptotic stability of uniformly bounded nonlinear switched systems
Affine-quadratic problems on Lie groups
1. | Department of Mathematics, University of Toronto, 40 St. George st, Toronto, Canada |
References:
[1] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View,", Encyclopedia of Mathematical Sciences, (2004).
|
[2] |
D. V. Anosov, A note on the Kepler problem,, Jour. Dynamical and Control Syst., 8 (2002), 413.
doi: 10.1023/A:1016386605889. |
[3] |
V. Arnold, "Les Méthodes Mathématiques de la Mécanique Classique,", Traduction francaise, (1974). Google Scholar |
[4] |
J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry,", Revised reprint of the 1975 original. AMS Chelsea Publishing, (1975).
|
[5] |
A. V. Bolsinov, A criterion for the completeness of a family of functions in involution that is constructed by the argument translation method,, (Russian) Dokl. Akad. Nauk SSSR, 301 (1988), 1037.
|
[6] |
A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces,, Math. Zeit., 246 (2004), 213.
doi: 10.1007/s00209-003-0596-x. |
[7] |
B. Bonnard, V. Jurdjevic, I. K. Kupka and G. Sallet, Transitivity of families of invariant vector fields on semi-direct products of lie groups,, Trans. Amer. Math. Soc., 271 (1982), 525.
doi: 10.1090/S0002-9947-1982-0654849-4. |
[8] |
P. Eberlein, "Geometry of Nonpositively Curved Manifolds,", Chicago Lectures in Mathematics. University of Chicago Press, (1996).
|
[9] |
S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces,", Pure and Applied Mathematics, (1978).
|
[10] |
Y. N. Fedorov and B. Jovanovic, Geodesic flows and newmann systems on steifel varieties,, Geometry and Integrability, (). Google Scholar |
[11] |
V. A. Fock, The hydrogen atom and non-euclidean geometry,, Izv. Akad. Nauk SSSR, (1935). Google Scholar |
[12] |
B. S. Goh, Necessary conditions for singular extremals involving multiple control variables,, SIAM J. Control, 4 (1966), 716.
doi: 10.1137/0304052. |
[13] |
P. Griffiths, "Exterior Differential Calculus and the Calculus of Variations,", Birkhauser, (1992). Google Scholar |
[14] |
C. G. J. Jacobi, "Vorlesungen Uber Dynamic,", Druck und Verlag von G. Reimer, (1884). Google Scholar |
[15] |
V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Advanced Mathematics 52, (1997).
|
[16] |
V. Jurdjevic, Hamiltonian point of view of non-euclidean geometry and elliptic functions,, System and Control Letters, 43 (2005), 25.
doi: 10.1016/S0167-6911(01)00093-7. |
[17] |
V. Jurdjevic, Optimal control, geometry and mechanics,, Mathematical Control Theory, (1999), 227.
|
[18] |
V. Jurdjevic and F. Monroy-Perez, Hamiltonian systems on Lie groups: Elastic curves, tops and constrained geodesic problems,, in, (2002), 3.
doi: 10.1142/9789812778079_0001. |
[19] |
V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems,, Regular and Chaotic Dyn., 16 (2011), 514.
doi: 10.1134/S156035471105008X. |
[20] |
V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).
doi: 10.1090/memo/0838. |
[21] |
V. Jurdjevic, The elliptic geodesic problem on the sphere,, in preparation., (). Google Scholar |
[22] |
H. Knorrer, Geodesics on quadrics and a mechanical problem of C. Newmann,, J. Reine Angew. Math., 334 (1982), 69.
doi: 10.1515/crll.1982.334.69. |
[23] |
J. Langer and D. Singer, Knotted elastic curves in $\mathbbR ^3$,, J. London. Math. Soc., 30 (1984), 512.
doi: 10.1112/jlms/s2-30.3.512. |
[24] |
P. Lee, "Kepler's Problem,", Master's thesis, (2004). Google Scholar |
[25] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure Appl. Math., 23 (1970), 609.
doi: 10.1002/cpa.3160230406. |
[26] |
J. Moser, Geometry of quadrics and spectral theory,, The Chern Symposium 1979 (Proc. Internat. Sympos., (1979), 147.
|
[27] |
J. Moser, "Integrable Hamiltonian Systems and Spectral Theory,", Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, (1983).
|
[28] |
C. Newmann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum,, J. Reine Angew. Math., 56 (1856), 345. Google Scholar |
[29] |
Y. Osipov, The Kepler problem and geodesic flows in spaces of constant curvature,, Celestial Mechanics, 16 (1977), 191.
doi: 10.1007/BF01228600. |
[30] |
A. M. Perelomov, "Integrable Systems Of Classical Mechanics And Lie Algebras, Vol. I,", Translated from the Russian by A. G. Reyman [A. G. Reiman]. Birkhauser Verlag, (1990).
doi: 10.1007/978-3-0348-9257-5. |
[31] |
T. Ratiu, The C. Newmann problem as a completely integrable system on an adjoint orbit,, Trans. Amer. Math. Soc., 264 (1981), 321.
doi: 10.2307/1998542. |
[32] |
A. G. Reyman, Integrable hamiltonian systems connected with graded Lie algebras,, J. Sov. Math., 19 (1982), 1507. Google Scholar |
[33] |
A. G. Reyman and S. Tian- Shansky, Group theoretic methods in the theory of finite dimensional integrable systems,, Encyclopaedia of Mathematical Sciences, (1994).
doi: 10.1007/978-3-642-57884-7. |
show all references
References:
[1] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View,", Encyclopedia of Mathematical Sciences, (2004).
|
[2] |
D. V. Anosov, A note on the Kepler problem,, Jour. Dynamical and Control Syst., 8 (2002), 413.
doi: 10.1023/A:1016386605889. |
[3] |
V. Arnold, "Les Méthodes Mathématiques de la Mécanique Classique,", Traduction francaise, (1974). Google Scholar |
[4] |
J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry,", Revised reprint of the 1975 original. AMS Chelsea Publishing, (1975).
|
[5] |
A. V. Bolsinov, A criterion for the completeness of a family of functions in involution that is constructed by the argument translation method,, (Russian) Dokl. Akad. Nauk SSSR, 301 (1988), 1037.
|
[6] |
A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces,, Math. Zeit., 246 (2004), 213.
doi: 10.1007/s00209-003-0596-x. |
[7] |
B. Bonnard, V. Jurdjevic, I. K. Kupka and G. Sallet, Transitivity of families of invariant vector fields on semi-direct products of lie groups,, Trans. Amer. Math. Soc., 271 (1982), 525.
doi: 10.1090/S0002-9947-1982-0654849-4. |
[8] |
P. Eberlein, "Geometry of Nonpositively Curved Manifolds,", Chicago Lectures in Mathematics. University of Chicago Press, (1996).
|
[9] |
S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces,", Pure and Applied Mathematics, (1978).
|
[10] |
Y. N. Fedorov and B. Jovanovic, Geodesic flows and newmann systems on steifel varieties,, Geometry and Integrability, (). Google Scholar |
[11] |
V. A. Fock, The hydrogen atom and non-euclidean geometry,, Izv. Akad. Nauk SSSR, (1935). Google Scholar |
[12] |
B. S. Goh, Necessary conditions for singular extremals involving multiple control variables,, SIAM J. Control, 4 (1966), 716.
doi: 10.1137/0304052. |
[13] |
P. Griffiths, "Exterior Differential Calculus and the Calculus of Variations,", Birkhauser, (1992). Google Scholar |
[14] |
C. G. J. Jacobi, "Vorlesungen Uber Dynamic,", Druck und Verlag von G. Reimer, (1884). Google Scholar |
[15] |
V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Advanced Mathematics 52, (1997).
|
[16] |
V. Jurdjevic, Hamiltonian point of view of non-euclidean geometry and elliptic functions,, System and Control Letters, 43 (2005), 25.
doi: 10.1016/S0167-6911(01)00093-7. |
[17] |
V. Jurdjevic, Optimal control, geometry and mechanics,, Mathematical Control Theory, (1999), 227.
|
[18] |
V. Jurdjevic and F. Monroy-Perez, Hamiltonian systems on Lie groups: Elastic curves, tops and constrained geodesic problems,, in, (2002), 3.
doi: 10.1142/9789812778079_0001. |
[19] |
V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems,, Regular and Chaotic Dyn., 16 (2011), 514.
doi: 10.1134/S156035471105008X. |
[20] |
V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005).
doi: 10.1090/memo/0838. |
[21] |
V. Jurdjevic, The elliptic geodesic problem on the sphere,, in preparation., (). Google Scholar |
[22] |
H. Knorrer, Geodesics on quadrics and a mechanical problem of C. Newmann,, J. Reine Angew. Math., 334 (1982), 69.
doi: 10.1515/crll.1982.334.69. |
[23] |
J. Langer and D. Singer, Knotted elastic curves in $\mathbbR ^3$,, J. London. Math. Soc., 30 (1984), 512.
doi: 10.1112/jlms/s2-30.3.512. |
[24] |
P. Lee, "Kepler's Problem,", Master's thesis, (2004). Google Scholar |
[25] |
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,, Comm. Pure Appl. Math., 23 (1970), 609.
doi: 10.1002/cpa.3160230406. |
[26] |
J. Moser, Geometry of quadrics and spectral theory,, The Chern Symposium 1979 (Proc. Internat. Sympos., (1979), 147.
|
[27] |
J. Moser, "Integrable Hamiltonian Systems and Spectral Theory,", Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, (1983).
|
[28] |
C. Newmann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum,, J. Reine Angew. Math., 56 (1856), 345. Google Scholar |
[29] |
Y. Osipov, The Kepler problem and geodesic flows in spaces of constant curvature,, Celestial Mechanics, 16 (1977), 191.
doi: 10.1007/BF01228600. |
[30] |
A. M. Perelomov, "Integrable Systems Of Classical Mechanics And Lie Algebras, Vol. I,", Translated from the Russian by A. G. Reyman [A. G. Reiman]. Birkhauser Verlag, (1990).
doi: 10.1007/978-3-0348-9257-5. |
[31] |
T. Ratiu, The C. Newmann problem as a completely integrable system on an adjoint orbit,, Trans. Amer. Math. Soc., 264 (1981), 321.
doi: 10.2307/1998542. |
[32] |
A. G. Reyman, Integrable hamiltonian systems connected with graded Lie algebras,, J. Sov. Math., 19 (1982), 1507. Google Scholar |
[33] |
A. G. Reyman and S. Tian- Shansky, Group theoretic methods in the theory of finite dimensional integrable systems,, Encyclopaedia of Mathematical Sciences, (1994).
doi: 10.1007/978-3-642-57884-7. |
[1] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[2] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[3] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[4] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[5] |
Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 |
[6] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[7] |
Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046 |
[8] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[9] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[10] |
Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 |
[11] |
Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124 |
[12] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 |
[13] |
Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 |
[14] |
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 |
[15] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
[16] |
Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144 |
[17] |
Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021003 |
[18] |
Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020347 |
[19] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[20] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]