Dynamic homotopy and landscape dynamical set topology in quantum control

Dominy, Jason; Rabitz, Herschel

doi:10.1063/1.4742375

Cited by 10 publications

(19 citation statements)

References 44 publications

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“…The key property for studying the topology of horizontal path spaces is the homotopy lifting property for the endpoint map. Our first result generalizes the main results from [12,21], proving that there exists p c > 1 (depending on F ) such that endpoint map is a Hurewicz fibration for the W 1,p topology for all 1 ≤ p < p c (i.e. F has the homotopy lifting property with respect to any space for these topologies).…”

Section: Introductionsupporting

confidence: 77%

“…Our proof of the previous theorem is much inspired from [21,12] and in fact consists in a simple (but important) modification of the proof from [21]. This theorem has two immediate consequences for the W 1,p topology (1 < p < p c ): (i) all the spaces Ω(y) as y varies on M are homotopy equivalent; (ii) the inclusion of Ω(y) into the ordinary space of curves (with no nonholonomic constraints) is a weak homotopy equivalence.…”

Section: Introductionmentioning

confidence: 99%

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Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry

Boarotto

Lerario

2017

Communications in Analysis and Geometry

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We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some W 1,p topology on the space of trajectories, for a certain p > 1. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the number of their critical points is infinite (we use Lusternik-Schnirelmann category combined with the Hurewicz property). In the special case where the control system is subriemannian this result can be read as the corresponding version of Serre's theorem, on the existence of infinitely many geodesics between two points on a compact riemannian manifold. In the subriemannian case we show that the Hurewicz property holds for all p ≥ 1 and the horizontal-loop space with the W 1,2 topology has the homotopy type of a CW-complex (as long as the endpoint map has at least one regular value); in particular the inclusion of the horizontal-loop space in the ordinary one is a homotopy equivalence.

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Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry

Boarotto

Lerario

2017

Communications in Analysis and Geometry

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“…Proof of Theorem 22. First of all observe that, since by assumption c α > 0 and (M, ∆) is contact, then every γ ∈ g −1 (c α ) is regular in the sense of (12). Note also that condition (18) in Proposition 15 is equivalent to ∇g = 0.…”

Section: 3mentioning

confidence: 92%

“…Remark 1. The uniform convergence topology on Ω has been studied in [25] and the W 1,1 in [12]. The W 1,p topology with p > 1 has been investigated by the first author and F. Boarotto in [8] -for the scopes of calculus of variations the case p > 1 is especially interesting as one can apply classical techniques from critical point theory to many problems of interest.…”

Section: Introductionmentioning

confidence: 99%

Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics

Lerario

Mondino

2019

Trans. Amer. Math. Soc. Ser. B

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Given a manifold M and a proper sub-bundle ∆ ⊂ T M , we investigate homotopy properties of the horizontal free loop space Λ, i.e. the space of absolutely continuous maps γ : S 1 → M whose velocities are constrained to ∆ (for example: legendrian knots in a contact manifold).In the first part of the paper we prove that the base-point map F : Λ → M (the map associating to every loop its base-point) is a Hurewicz fibration for the W 1,2 topology on Λ. Using this result we show that, even if the space Λ might have deep singularities (for example: constant loops form a singular manifold homeomorphic to M ), its homotopy can be controlled nicely. In particular we prove that Λ (with the W 1,2 topology) has the homotopy type of a CW-complex, that its inclusion in the standard free loop space (i.e. the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently its homotopy groups can be computed as π k (Λ) ≃ π k (M ) ⋉ π k+1 (M ) for all k ≥ 0.In the second part of the paper we address the problem of the existence of closed subriemannian geodesics. In the general case we prove that if (M, ∆) is a compact sub-riemannian manifold, each non trivial homotopy class in π 1 (M ) can be represented by a closed subriemannian geodesic.In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if (M, ∆) is a compact, contact manifold, then every sub-riemannian metric on ∆ carries at least one closed sub-riemannian geodesic. This result is based on a combination of the above topological results with the delicate study of an analogous of a Palais-Smale condition in the vicinity of abnormal loops (singular points of Λ).

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“…Remark 1. The study of the space of maps with some restrictions on their differential goes back to the works on immersions of S. Smale [16]; the case of trajectories of affine control systems were studied first by A. V. Sarychev [14] for the uniform convergence topology, by J. Dominy and H. Rabitz [10] for the W 1,1 topology and by the last two authors of this paper for the W 1,p , p > 1 topology [3]. The case of closed W 1,2 curves on nonholonomic distribution has been addressed by the last author and A. Mondino on [9].…”

mentioning

confidence: 99%

Homotopically invisible singular curves

2017

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Abstract. Given a smooth manifold M and a totally nonholonomic distribution ∆ ⊂ T M of rank d, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M . Singular curves are critical points of the endpoint map F : γ → γ(1) defined on the space Ω of horizontal paths starting at a fixed point x. We consider a subriemannian energy J : Ω(y) → R, where Ω(y) = F −1 (y) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets {γ ∈ Ω(y) | J(γ) ≤ E}. We call them homotopically invisible. It turns out that for d ≥ 3 generic subriemannian structures in the sense of [5] have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a subriemannian Minimax principle and discuss some applications).

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Dynamic homotopy and landscape dynamical set topology in quantum control

Cited by 10 publications

References 44 publications

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry

Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics

Homotopically invisible singular curves

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