no. 2
Optimal control
Pointwise second-order necessary conditions for optimal control problems evolved on Riemannian manifolds
[Conditions nécessaires ponctuelles du second ordre pour des problèmes de contrôle optimal évolués sur une variété riemannienne]

Présenté par : Jean-Michel Coron

Cui, Qing1 ; Deng, Li1 ; Zhang, Xu2
1 School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan Province, China
2 School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 191-194.

Dans cette Note, nous étudions un problème du contrôle optimal sur une variété riemannienne. Dans ce problème, l'ensemble des contrôles est un espace de Polish général ; ainsi, la technique de variation classique ne s'applique pas ici. On obtient une condition d'optimalité ponctuelle du second ordre, pour laquelle le tenseur de courbure de la variété apparaît explicitement dans l'équation duale du second ordre.

In this Note, we study an optimal control problem on a Riemannian manifold. The control set in our problem is assumed to be a general Polish space, and therefore the classical variation technique fails. We obtain a pointwise second-order optimality condition, for which the curvature tensor of the manifold appears explicitly in the second-order dual equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.09.032
Cui, Qing 1 ; Deng, Li 1 ; Zhang, Xu 2

1 School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan Province, China
2 School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China 
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Cui, Qing; Deng, Li; Zhang, Xu. Pointwise second-order necessary conditions for optimal control problems evolved on Riemannian manifolds. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 191-194. doi : 10.1016/j.crma.2015.09.032. http://www.numdam.org/articles/10.1016/j.crma.2015.09.032/

[1] Agrachev, A.A.; Sachkov, Y.L. Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., vol. 87, Springer-Verlag, Berlin, 2004

[2] Bell, D.J.; Jacobson, D.H. Singular Optimal Control Problems, Academic Press, London, New York, 1975

[3] Bonnard, B.; Caillau, J.B.; Trélat, E. Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. Calc. Var., Volume 13 (2007), pp. 207-236

[4] Clements, D.J.; Anderson, B.D.O. Singular Optimal Control: The Linear-Quadratic Problem, Springer-Verlag, Berlin, New York, 1978

[5] Cui, Q.; Deng, L.; Zhang, X. Second order necessary conditions for optimal control problems on Riemannian manifolds | arXiv

[6] Frankowska, H.; Tonon, D. Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints, SIAM J. Control Optim., Volume 51 (2013), pp. 3814-3843

[7] Gabasov, R.F.; Kirillova, F.M. High order necessary conditions for optimality, SIAM J. Control, Volume 10 (1972), pp. 127-168

[8] Goh, B.S. Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, Volume 4 (1966), pp. 716-731

[9] Knobloch, H.-W. Higher Order Necessary Conditions in Optimal Control Theory, Springer-Verlag, Berlin, New York, 1981

[10] Krener, A.J. The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., Volume 15 (1977), pp. 256-293

[11] Lou, H. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure, Discrete Contin. Dyn. Syst., Ser. B, Volume 14 (2010), pp. 1445-1464

[12] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mischenko, E.F. Mathematical Theory of Optimal Processes, Wiley, New York, 1962

[13] Schättler, H.; Ledzewicz, U. Geometric Optimal Control, Theory, Methods and Examples, Interdiscip. Appl. Math., vol. 38, Springer, New York, 2012

[14] Bonnans, J.F.; Hermant, A. Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 561-598

[15] Hoehener, D. Variational approach to second-order optimality conditions for control problems with pure state constraints, SIAM J. Control Optim., Volume 50 (2012), pp. 1139-1173

[16] Páles, Z.; Zeidan, V. Optimal control problems with set-valued control and state constraints, SIAM J. Optim., Volume 14 (2003), pp. 334-358

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