Singular Trajectories of Control-Affine Systems
Singular Trajectories of Control-Affine Systems
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控制仿射系统的奇异轨迹
DOI：
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发表时间：
2006
期刊：
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通讯作者：
E. Trélat
中科院分区：
文献类型：
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作者：
Y. Chitour;F. Jean;E. Trélat
当将最优控制方法应用于运动规划或稳定问题时，我们看到由于特定轨迹的存在，即最小化潜在最优控制问题的奇异轨迹，可能会出现一些理论或数值困难。在本文中，我们提供了控制仿射系统的奇异轨迹的表征。我们证明，在一般假设下，这些轨迹具有与计算方面相关的良好特性；更准确地说，我们证明了，对于一个一般系统——关于Whitney拓扑——所有的非平凡奇异轨迹都是最小阶的，并且是corank 1的。这些结果，建立在无漂移和控制仿射系统，推广了[Y]。Chitour， F. Jean和E. Trelat，《计算机数学》。， 337(2003)，第49-52页（法文）；刘建军，刘建军，李建军，等。， 73 (2006), pp. 45-73]。因此，对于由两个以上矢量场定义的一般控制仿射系统（有或没有漂移），并且对于固定成本，不存在最小化奇异轨迹。此外，我们证明了给定一个满足李代数秩条件（LARC）的控制仿射系统，奇异轨迹在代价方面是严格异常的。然后，我们展示了如何使用这些结果来推导值函数和汉密尔顿-雅可比方程理论的规律性结果，从理论和实现的角度来看，这些结果反过来又应用于稳定和运动规划。
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.