Affine-quadratic problems on Lie groups

Velimir Jurdjevic

doi:10.3934/mcrf.2013.3.347

Previous Article
MCRF Home
This Issue
Next Article
Asymptotic stability of uniformly bounded nonlinear switched systems

September 2013, 3(3): 347-374. doi: 10.3934/mcrf.2013.3.347

Affine-quadratic problems on Lie groups

Velimir Jurdjevic ^1,

1.	Department of Mathematics, University of Toronto, 40 St. George st, Toronto, Canada

Received December 2012 Revised March 2013 Published September 2013

Abstract
Related Papers

This paper focuses on a class of left invariant variational problems on a Lie group$\ G\ $whose Lie algebra $\mathfrak{g}$ admits Cartan decomposition $ \mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with the usual Lie algebraic conditions \begin{equation*} \lbrack \mathfrak{p},\mathfrak{p]\subseteq k\ },\ \mathfrak{[p},\mathfrak{ k]\subseteq p},\mathfrak{\ [k},\mathfrak{k]\subseteq k.} \end{equation*} The Maximum Principle of optimal control leads to the Hamiltonians $H$ on $ \mathfrak{g\ }$that admit spectral parameter representations with important contributions to the theory of integrable Hamiltonian systems. Particular cases will be singled out that provides natural explanations for the classical results of Fock and Moser linking Kepler's problem to the geodesics on spaces of constant curvature, C.L. Jacobi's geodesic problem on an ellipsoid and J.Moser's work on integrability based on isospectral methods. The paper also shows the relevance of this class of Hamiltonians to the elastic curves on spaces of constant curvature.

Keywords: Lie group., Affine-quadratic problem, optimal control, Hamiltonian system.

Mathematics Subject Classification: Primary 37K05, 49J15 ; Secondary 14H7.

Citation: Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347

References:

[1]	A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View,", Encyclopedia of Mathematical Sciences, (2004). Google Scholar
[2]	D. V. Anosov, A note on the Kepler problem,, Jour. Dynamical and Control Syst., 8 (2002), 413. doi: 10.1023/A:1016386605889. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	P. Eberlein, "Geometry of Nonpositively Curved Manifolds,", Chicago Lectures in Mathematics. University of Chicago Press, (1996). Google Scholar
[9]	S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces,", Pure and Applied Mathematics, (1978). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[17]	V. Jurdjevic, Optimal control, geometry and mechanics,, Mathematical Control Theory, (1999), 227. Google Scholar
[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. Google Scholar
[19]	V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems,, Regular and Chaotic Dyn., 16 (2011), 514. doi: 10.1134/S156035471105008X. Google Scholar
[20]	V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005). doi: 10.1090/memo/0838. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. Moser, Geometry of quadrics and spectral theory,, The Chern Symposium 1979 (Proc. Internat. Sympos., (1979), 147. Google Scholar
[27]	J. Moser, "Integrable Hamiltonian Systems and Spectral Theory,", Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, (1983). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View,", Encyclopedia of Mathematical Sciences, (2004). Google Scholar
[2]	D. V. Anosov, A note on the Kepler problem,, Jour. Dynamical and Control Syst., 8 (2002), 413. doi: 10.1023/A:1016386605889. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	P. Eberlein, "Geometry of Nonpositively Curved Manifolds,", Chicago Lectures in Mathematics. University of Chicago Press, (1996). Google Scholar
[9]	S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces,", Pure and Applied Mathematics, (1978). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[17]	V. Jurdjevic, Optimal control, geometry and mechanics,, Mathematical Control Theory, (1999), 227. Google Scholar
[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. Google Scholar
[19]	V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems,, Regular and Chaotic Dyn., 16 (2011), 514. doi: 10.1134/S156035471105008X. Google Scholar
[20]	V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups,, Mem. Amer. Math. Soc., 178 (2005). doi: 10.1090/memo/0838. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	J. Moser, Geometry of quadrics and spectral theory,, The Chern Symposium 1979 (Proc. Internat. Sympos., (1979), 147. Google Scholar
[27]	J. Moser, "Integrable Hamiltonian Systems and Spectral Theory,", Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, (1983). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
[2]	Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737
[3]	Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83
[4]	Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
[5]	Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[6]	Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105
[7]	Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967
[8]	Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[9]	Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451
[10]	Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197
[11]	Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.
[12]	Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261
[13]	Ben Muatjetjeja, Dimpho Millicent Mothibi, Chaudry Masood Khalique. Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2803-2812. doi: 10.3934/dcdss.2020219
[14]	Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[15]	Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651
[16]	Gülden Gün Polat, Teoman Özer. On group analysis of optimal control problems in economic growth models. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2853-2876. doi: 10.3934/dcdss.2020215
[17]	Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
[18]	Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[19]	Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016
[20]	Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

2019 Impact Factor: 0.857

American Institute of Mathematical Sciences

Affine-quadratic problems on Lie groups

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Affine-quadratic problems on Lie groups

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors