Singular Trajectories of Control-Affine Systems

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 system—with respect to the Whitney topology—all 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. Trelat, Comptes Rendus Math., 337 (2003), pp. 49-52 (in French); Y. Chitour, F. Jean, and E. Trelat, J. Differential Geom., 73 (2006), pp. 45-73]. 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 Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, from both theoretical and implementational points of view.