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• W2108551874 abstract "For ¿ ¿ (0, ¿/2), let (¿) <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>¿</sub> be the control system x¿ = (F + uG)x, where x belongs to the two-dimensional unit sphere S <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> , u ¿ [-1, 1] and F,G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms cos(¿) and sin(¿). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0, 1) <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>T</sup> associated to (¿) <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>¿</sub> , as the parameter ¿ tends to zero; this problem is motivated by specific issues in the control of two-level quantum systems subject to weak external fields. The TOS is characterized by a ¿two-snakes¿ configuration on the whole S <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> , except for a neighborhood U <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>¿</sub> of the south pole (0, 0,-1) <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>T</sup> of diameter at most O(¿). Inside U <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>¿</sub> , the TOS depends on the relationship between r(¿) := ¿/2¿-[¿/2¿] and ¿. More precisely, we characterize three main relationships, by considering sequences (¿ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> ) <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k¿0</sub> satisfying (a) r(¿ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> ) = r¿ (b) r(¿ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> ) = C¿ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> and (c) r(¿ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> ) = 0, where r¿ ¿ (0, 1) and C ¿ 0. In each case we describe the TOS and, in the case (a), we provide, after a suitable rescaling, the limit behavior of the corresponding TOS inside U <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>¿</sub> , as ¿ tends to zero." @default.
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• W2108551874 date "2008-12-01" @default.
• W2108551874 modified "2023-10-14" @default.
• W2108551874 title "Limit time optimal synthesis for a two-level quantum system" @default.
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• W2108551874 doi "https://doi.org/10.1109/cdc.2008.4738782" @default.
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