Abstract
Let \(\dot q = f(q) + ug(q)\) be a smooth control system on a three-dimensional manifold. Given a point q 0 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 q 0. 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.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.
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Sigalotti, M. Regularity properties of optimal trajectories of single-input control systems in dimension three. J Math Sci 126, 1561–1573 (2005). https://doi.org/10.1007/s10958-005-0044-z
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DOI: https://doi.org/10.1007/s10958-005-0044-z