Regularity properties of optimal trajectories of single-input control systems in dimension three
Let \(\dot q = f(q) + ug(q)\) be a smooth control system on a three-dimensional manifold. Given a point q0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.
KeywordsManifold Switching Time Optimal Trajectory Regularity Property Independence Condition
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