Singular Trajectories of Control-Affine Systems

Abstract

Abstract. When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing 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 [13,14]. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the 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, both from the theoretical and implementation issues.1. Introduction. When addressing standard issues of control theory such as motion planning and stabilization, one may adopt an approach based on optimal control, e.g., Hamilton-Jacobi type methods and shooting algorithms. One is then immediately facing intrinsic difficulties due to the possible presence of singular trajectories. It is therefore important to characterize these trajectories, by studying in particular their existence, optimality status, and the related computational aspects. In this paper, we provide answers to the aforementioned questions for control-affine systems, under generic assumptions, and then investigate consequences in optimal control and its applications.Let M be a smooth (i.e. C ∞ ) manifold of dimension n. Consider the control-affine system

Optimal Control and Applications to Aerospace: Some Results and Challenges

“…We refer the reader to [19] for more details on this theory. It can also be shown that this kind of computation is valid in a "generic" situation (see [44,45,46,47]). …”

“…of [68,62,69,65,66,77,67] that are applicable to bang-bang situations in any dimension should permit to derive local optimal syntheses in larger dimension under additional assumptions, and as well for control-affine systems with more than one control (although it can be expected that the situation is much more complicated). Note however that, according to the results of [45,46,47], generic (in the Whitney sense) controlaffine systems do not admit any minimizing singular trajectory whenever the number of controls is more than two (more precisely it is shown in these references that such generic control-affine systems do not admit any trajectories satisfying the Goh necessary condition derived in [160]). For a first result concerning the classification of extremals for control-affine systems with two controls, we quote the recent article [139], with an application to the minimum time control of the restricted three-body problem.…”

Mass Transportation on Sub-Riemannian Manifolds

“…Let us moreover observe that, given a medium-fat distribution, it can be shown that for a generic smooth complete Riemannian metric on M the distribution does not admit nontrivial singular sub-Riemannian minimizing geodesics (see [15,16]). As a consequence, we have: Proposition 4.7.…”

“…Denote by D m the space of rank m distributions on M endowed with the Whitney C ∞ topology. Chitour, Jean and Trélat proved that there exists an open dense subset O m of D m such that every element of O m does not admit nontrivial minimizing singular paths (see [15,16]). As a consequence, we have:…”

Geometric Approach to Pontryagin’s Maximum Principle

“…4. That is why the existence or non-existence of abnormal minimizers is only known for specific control problems, mainly control-linear and control-affine systems with control-quadratic cost functions or for time-optimal control problems [4,6,19,26,27,77].…”