Limit Time Optimal Synthesis for a Control-Affine System on S²

For α ∈]0, π/2[, let (Σ)α be the control system ˙x = (F + uG)x, where x belongs to the two-dimensional unit sphere S 2, u ∈ [−1, 1] and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes of respective length cos(α) and sin(α). In this paper, we study the time optimal...

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Bibliographic Details
Main Authors: P. Mason, R. Salmoni, U. Boscain, Y. Chitour
Other Authors: The Pennsylvania State University CiteSeerX Archives
Format: Text
Language:English
Published: 2008
Subjects:
control-affine systems
optimal synthesis
minimum time
asymptotics
South Pole
North Pole
South pole
Online Access:http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.95
http://arxiv.org/pdf/math/0611823v1.pdf
Description
Summary:For α ∈]0, π/2[, let (Σ)α be the control system ˙x = (F + uG)x, where x belongs to the two-dimensional unit sphere S 2, u ∈ [−1, 1] and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes of respective length cos(α) and sin(α). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0, 1) T associated to (Σ)α, as the parameter α tends to zero. We first prove that the TOS is characterized by a “two-snakes ” configuration on the whole S 2, except for a neighborhood Uα of the south pole (0, 0, −1) T of diameter at most O(α). We next show that, inside Uα, the TOS depends on the relationship between r(α): = π/2α −[π/2α] and α. More precisely, we characterize three main relationships, by considering sequences (αk)k≥0

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