Limit time optimal synthesis for a control-affine system on S2
International audience For α ∈]0,π/2[, let (Σ)α be the control system ẋ = (F+uG)x, where x belongs to the two-dimensional unit sphere S2, u ∈ [-1, 1], and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms cos(α) and sin(α). In this paper, we...
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ftuniparissaclay:oai:HAL:hal-02320804v1 2024-04-28T08:31:59+00:00 Limit time optimal synthesis for a control-affine system on S2 Mason, Paolo Salmoni, Rebecca Boscain, Ugo Chitour, Yacine Institut Élie Cartan de Nancy (IECN) Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS) Laboratoire des signaux et systèmes (L2S) Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS) SISSA MathLab Trieste 2008 https://hal.science/hal-02320804 https://doi.org/10.1137/060675988 en eng HAL CCSD Society for Industrial and Applied Mathematics info:eu-repo/semantics/altIdentifier/doi/10.1137/060675988 hal-02320804 https://hal.science/hal-02320804 doi:10.1137/060675988 ISSN: 0363-0129 EISSN: 1095-7138 SIAM Journal on Control and Optimization https://hal.science/hal-02320804 SIAM Journal on Control and Optimization, 2008, 47 (1), pp.111-143. ⟨10.1137/060675988⟩ Asymptotics Control of quantum systems Control-affine systems Minimum time Optimal synthesis [MATH]Mathematics [math] info:eu-repo/semantics/article Journal articles 2008 ftuniparissaclay https://doi.org/10.1137/060675988 2024-04-08T17:41:42Z International audience For α ∈]0,π/2[, let (Σ)α be the control system ẋ = (F+uG)x, where x belongs to the two-dimensional unit sphere S2, u ∈ [-1, 1], and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms 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 a tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a "two-snakes" configuration on the whole S2, except for a neighborhood Uα of the south pole (0, 0, -1)T of diameter at most Ο(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 satisfying (a) r(αk) = r̄, (b) r(αk) = Cαk, and (c) r(αk) = 0, where r̄ ∈ (0, 1) and C andgt; 0. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as α tends to zero, of the corresponding TOS inside Uα. © 2008 Society for Industrial and Applied Mathematics. Article in Journal/Newspaper North Pole South pole Archives ouvertes de Paris-Saclay SIAM Journal on Control and Optimization 47 1 111 143 |
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Archives ouvertes de Paris-Saclay |
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Asymptotics Control of quantum systems Control-affine systems Minimum time Optimal synthesis [MATH]Mathematics [math] |
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Asymptotics Control of quantum systems Control-affine systems Minimum time Optimal synthesis [MATH]Mathematics [math] Mason, Paolo Salmoni, Rebecca Boscain, Ugo Chitour, Yacine Limit time optimal synthesis for a control-affine system on S2 |
topic_facet |
Asymptotics Control of quantum systems Control-affine systems Minimum time Optimal synthesis [MATH]Mathematics [math] |
description |
International audience For α ∈]0,π/2[, let (Σ)α be the control system ẋ = (F+uG)x, where x belongs to the two-dimensional unit sphere S2, u ∈ [-1, 1], and F, G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms 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 a tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a "two-snakes" configuration on the whole S2, except for a neighborhood Uα of the south pole (0, 0, -1)T of diameter at most Ο(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 satisfying (a) r(αk) = r̄, (b) r(αk) = Cαk, and (c) r(αk) = 0, where r̄ ∈ (0, 1) and C andgt; 0. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as α tends to zero, of the corresponding TOS inside Uα. © 2008 Society for Industrial and Applied Mathematics. |
author2 |
Institut Élie Cartan de Nancy (IECN) Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS) Laboratoire des signaux et systèmes (L2S) Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS) SISSA MathLab Trieste |
format |
Article in Journal/Newspaper |
author |
Mason, Paolo Salmoni, Rebecca Boscain, Ugo Chitour, Yacine |
author_facet |
Mason, Paolo Salmoni, Rebecca Boscain, Ugo Chitour, Yacine |
author_sort |
Mason, Paolo |
title |
Limit time optimal synthesis for a control-affine system on S2 |
title_short |
Limit time optimal synthesis for a control-affine system on S2 |
title_full |
Limit time optimal synthesis for a control-affine system on S2 |
title_fullStr |
Limit time optimal synthesis for a control-affine system on S2 |
title_full_unstemmed |
Limit time optimal synthesis for a control-affine system on S2 |
title_sort |
limit time optimal synthesis for a control-affine system on s2 |
publisher |
HAL CCSD |
publishDate |
2008 |
url |
https://hal.science/hal-02320804 https://doi.org/10.1137/060675988 |
genre |
North Pole South pole |
genre_facet |
North Pole South pole |
op_source |
ISSN: 0363-0129 EISSN: 1095-7138 SIAM Journal on Control and Optimization https://hal.science/hal-02320804 SIAM Journal on Control and Optimization, 2008, 47 (1), pp.111-143. ⟨10.1137/060675988⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/doi/10.1137/060675988 hal-02320804 https://hal.science/hal-02320804 doi:10.1137/060675988 |
op_doi |
https://doi.org/10.1137/060675988 |
container_title |
SIAM Journal on Control and Optimization |
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47 |
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1 |
container_start_page |
111 |
op_container_end_page |
143 |
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1797589313399029760 |