NDLI: Singular Trajectories of Control-Affine Systems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chitour, Yacine Jean, Frdric Trlat, Emmanuel
Copyright Year	2008
Abstract	When applying methods of optimal control to motion planning or stabilization problems, we see that some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, minimizing singular trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic systemwith respect to the Whitney topologyall nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend results of [Y. Chitour, F. Jean, and E. Trlat, Comptes Rendus Math., 337 (2003), pp. 4952 (in French); Y. Chitour, F. Jean, and E. Trlat, J. Differential Geom., 73 (2006), pp. 4573]. As a consequence, for generic control-affine systems (with or without drift) defined by more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. Besides, we prove that, given a control-affine system satisfying the Lie algebra rank condition (LARC), singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of HamiltonJacobi equations, which in turn have applications for stabilization and motion planning, from both theoretical and implementational points of view.
Starting Page	1078
Ending Page	1095
Page Count	18
File Format	PDF
ISSN	03630129
DOI	10.1137/060663003
e-ISSN	10957138
Issue Number	2
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-03-19
Access Restriction	Subscribed
Subject Keyword	control-affine system Generic properties, structural stability genericity singular trajectory Optimal control problems involving ordinary differential equations optimal control
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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