This work is concerned with optimal control problems on Riemannian manifolds, for which two typical cases are considered. The first case is when the endpoint is free. For this case, the control set is assumed to be a separable metric space. By introducing suitable dual equations, which depend on the curvature tensor of the manifold, we establish the second order necessary and sufficient optimality conditions of integral form. Particularly, when the control set is a Polish space, the second order necessary condition is reduced to a pointwise form. As a key preliminary result and also an interesting byproduct, we derive a geometric lemma, which may have some independent interest. The second case is when the endpoint is fixed. For this more difficult case, the control set is assumed to be open in a Euclidian space, and we obtain the second order necessary and sufficient optimality conditions, in which the curvature tensor also appears explicitly. Our optimality conditions can be used to recover the following famous geometry result: the shortest geodesic connecting two fixed points on a Riemannian manifold satisfies the second variation of energy; while the existing optimality conditions in control literatures fail to give the same result.
Accepté le :
DOI : 10.1051/cocv/2018028
Mots-clés : Optimal control, second order necessary and sufficient conditions, Riemannian manifold, curvature tensor
@article{COCV_2019__25__A35_0,
author = {Cui, Qing and Deng, Li and Zhang, Xu},
title = {Second order optimality conditions for optimal control problems on {Riemannian} manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018028},
zbl = {1444.49012},
mrnumber = {4003463},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018028/}
}
TY - JOUR AU - Cui, Qing AU - Deng, Li AU - Zhang, Xu TI - Second order optimality conditions for optimal control problems on Riemannian manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018028/ DO - 10.1051/cocv/2018028 LA - en ID - COCV_2019__25__A35_0 ER -
%0 Journal Article %A Cui, Qing %A Deng, Li %A Zhang, Xu %T Second order optimality conditions for optimal control problems on Riemannian manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018028/ %R 10.1051/cocv/2018028 %G en %F COCV_2019__25__A35_0
Cui, Qing; Deng, Li; Zhang, Xu. Second order optimality conditions for optimal control problems on Riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 35. doi : 10.1051/cocv/2018028. http://www.numdam.org/articles/10.1051/cocv/2018028/
[1] and , Feedback-invariant optimal control theory and differential geometry. I. Regular extremals. J. Dynam. Control Syst. 3 (1997) 343–389. | DOI | MR | Zbl
[2] and , Symplectic geometry for optimal control, in Nonlinear Controllability and Optimal Control. Vol. 133 of Monographs and Textbooks in Pure and Applied Mathematics. Dekker, New York (1990) 263–277. | MR | Zbl
[3] and , Control Theory from the Geometric Viewpoint. Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). | MR | Zbl
[4] and , Singular Optimal Control Problems. Academic Press, London, New York (1975). | MR | Zbl
[5] and , Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 561–598. | DOI | Numdam | MR | Zbl
[6] and , No-gap second-order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. Ser. B 117 (2009) 21–50. | DOI | MR | Zbl
[7] and , Singular Trajectories and Their Role in Control Theory. Vol. 40 of Mathématiques & Applications. Springer-Verlag, Berlin, Heidelberg, New York (2003). | MR | Zbl
[8] , and , Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV 13 (2007) 207–236. | Numdam | MR | Zbl
[9] and , Optimal Syntheses for Control Systems on 2-D Manifolds. Vol. 43 of Mathématiques & Applications. Springer-Verlag, Berlin, Heidelberg, New York (2004). | MR | Zbl
[10] and , Singular Optimal Control: The Linear-Quadratic Problem. Springer-Verlag, Berlin, New York (1978). | MR | Zbl
[11] , and , Pointwise second order necessary conditions for optimal control problems evolved on Riemannian manifolds. C. R. Math. Acad. Sci. Paris Ser. I 354 (2016) 191–194. | DOI | MR | Zbl
[12] , Dynamic programming method for control systems on manifolds and its relations to maximum principle. J. Math. Anal. Appl. 434 (2016) 915–938. | DOI | MR | Zbl
[13] Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1976). | MR | Zbl
[14] , Riemannian Geometry, Translated from the Second Portuguese Edition by F. Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA (1992). | MR | Zbl
[15] and , Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints SIAM J. Control Optim. 51 (2013) 3814–3843. | DOI | MR | Zbl
[16] and , High order necessary conditions for optimality. SIAM J. Control 10 (1972) 127–168. | DOI | MR | Zbl
[17] and , Singular Optimal Controls. Izdat. “Nauka”, Moscow (1973). | MR
[18] , The second variation for the singular Bolza problem. SIAM J. Control 4 (1966) 309–325. | DOI | MR | Zbl
[19] , Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 (1966) 716–731. | DOI | MR | Zbl
[20] , Differential Geometry and Symmetric Spaces. Academic Press, New York, London (1962). | MR | Zbl
[21] , Variational approach to second-order optimality conditions for control problems with pure state constraints SIAM. J.Control Optim. 50 (2012) 1139–1173. | DOI | MR | Zbl
[22] , Variational approach to second-order sufficient optimality conditions in optimal control. SIAM. J. Control Optim. 52 (2014) 861–892. | DOI | MR | Zbl
[23] , Higher Order Necessary Conditions in Optimal Control Theory. Springer-Verlag, Berlin, New York (1981). | DOI | MR | Zbl
[24] and , Foundations of Differential Geometry, Vol. 1. Interscience, New York, London (1963). | MR
[25] , The high order maximal principle and its application to singular extremals, SIAM J. Control Optim. 15 (1977) 256–293. | DOI | MR | Zbl
[26] , Riemannian Manifolds: An Introduction to Curvature. Vol. 176 of Graduate Texts in Mathematics. Springer-Verlag, New York (1997). | DOI | MR | Zbl
[27] and , Optimal Control Theory for Infinite-Dimensional Systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1995). | MR | Zbl
[28] , Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1445–1464. | MR | Zbl
[29] , Sufficient optimality conditions for optimal control subject to state constraints. SIAM J. Control Optim. 35 (1997) 205–227. | DOI | MR | Zbl
[30] , and , Second-order sufficient conditions for state-constrained optimal control problems. J. Optim. Theory Appl. 123 (2004) 595–617. | DOI | MR | Zbl
[31] and , Applications to Regular and Bang-Bang Control. Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM, Philadelphia, PA (2012). | MR | Zbl
[32] and , Optimal control problems with set-valued control and state constraints. SIAM J. Optim. 14 (2003) 334–358. | DOI | MR | Zbl
[33] , Riemannian Geometry, 2nd edition. Vol. 171 of Graduate Texts in Mathematics. Springer-Verlag, New York (2006). | DOI | MR | Zbl
[34] , , and , Mathematical Theory of Optimal Processes. Wiley, New York (1962). | MR
[35] and , Geometric Optimal Control, Theory, Methods and Examples. Vol. 38 of Interdisciplinary Applied Mathematics. Springer, New York (2012). | DOI | MR | Zbl
[36] , Geometry and optimal control, in Mathematical Control Theory. Springer, New York (1999) 140–198. | DOI | MR | Zbl
[37] , A second-order condition that strengthens Pontryagin’s maximum principle. J. Differ. Equ. 28 (1978) 284–307. | DOI | MR | Zbl
[38] , and , An Introduction to Riemannian Geometry (in Chinese). Press of Peking University, Beijing (1989).
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