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Enrico Le Donne - One of the best experts on this subject based on the ideXlab platform.

  • Universal differentiability sets and maximal directional derivatives in Carnot groups
    2020
    Co-Authors: Enrico Le Donne, Pinamonti Andrea, Speight Gareth
    Abstract:

    We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz Map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.Comment: 30 page

  • Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
    2019
    Co-Authors: Antonelli Gioacchino, Enrico Le Donne
    Abstract:

    This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mathcal{H}^{12}$. As a consequence, we show that for every Carnot group of Hausdorff dimension 12, any Lipschitz Map defined on a subset of it with values in $S$ has $\mathcal{H}^{12}$-null image. In particular, we deduce that this smooth hypersurface cannot be Lipschitz parametrizable by countably many Maps each defined on some subset of some Carnot group of Hausdorff dimension $12$. As main consequence we have that a notion of rectifiability proposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioni and F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset $U$ of a homogeneous subgroup of Hausdorff dimension $12$ of a Carnot group, every bi-Lipschitz Map $f:U\to S$ satisfies $\mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{\infty}$-hypersurfaces in $\mathbb H^n$ with $n\geq 2$ are countably $\mathbb{H}^{n-1}\times\mathbb R$-rectifiabile according to Pauls' definition, even with bi-Lipschitz Maps.Comment: 30 page

  • universal differentiability sets and maximal directional derivatives in carnot groups
    arXiv: Functional Analysis, 2017
    Co-Authors: Enrico Le Donne, Andrea Pinamonti, Gareth Speight
    Abstract:

    We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz Map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.

Prats Martí - One of the best experts on this subject based on the ideXlab platform.

Prieur Christophe - One of the best experts on this subject based on the ideXlab platform.

  • Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Tanwani Aneel, Brogliato Bernard, Prieur Christophe
    Abstract:

    International audienceA class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued Mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz Map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued Mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available; In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters

  • Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities
    'Society for Industrial & Applied Mathematics (SIAM)', 2018
    Co-Authors: Tanwani Aneel, Brogliato Bernard, Prieur Christophe
    Abstract:

    A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued Mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz Map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued Mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available, In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters.Comment: Accepted for publication in SICON, SIAM Journal of Control and Optimization, 2018. Some technical arguments in the proofs from earlier version have been correcte

Le Donne Enrico - One of the best experts on this subject based on the ideXlab platform.

  • Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
    'Elsevier BV', 2020
    Co-Authors: Antonelli Gioacchino, Le Donne Enrico
    Abstract:

    This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz Map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in , has negligible image with respect to the Hausdorff measure . In particular, we deduce that cannot be Lipschitz parametrizable by countably many Maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz Map satisfies . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all -hypersurfaces in with are countably -rectifiable according to Pauls’ definition, even with bi-Lipschitz Maps.peerReviewe

  • Universal differentiability sets and maximal directional derivatives in Carnot groups
    'Elsevier BV', 2019
    Co-Authors: Le Donne Enrico, Pinamonti Andrea, Speight Gareth
    Abstract:

    We show that every Carnot group G of step 2 admits a Hausdorff dimension one ‘universal differentiability set’ N such that every Lipschitz Map f:G→R is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3

Antonelli Gioacchino - One of the best experts on this subject based on the ideXlab platform.

  • Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
    'Elsevier BV', 2020
    Co-Authors: Antonelli Gioacchino, Le Donne Enrico
    Abstract:

    This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz Map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in , has negligible image with respect to the Hausdorff measure . In particular, we deduce that cannot be Lipschitz parametrizable by countably many Maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz Map satisfies . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all -hypersurfaces in with are countably -rectifiable according to Pauls’ definition, even with bi-Lipschitz Maps.peerReviewe

  • Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
    2019
    Co-Authors: Antonelli Gioacchino, Enrico Le Donne
    Abstract:

    This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a $C^{\infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $\mathcal{H}^{12}$. As a consequence, we show that for every Carnot group of Hausdorff dimension 12, any Lipschitz Map defined on a subset of it with values in $S$ has $\mathcal{H}^{12}$-null image. In particular, we deduce that this smooth hypersurface cannot be Lipschitz parametrizable by countably many Maps each defined on some subset of some Carnot group of Hausdorff dimension $12$. As main consequence we have that a notion of rectifiability proposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioni and F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset $U$ of a homogeneous subgroup of Hausdorff dimension $12$ of a Carnot group, every bi-Lipschitz Map $f:U\to S$ satisfies $\mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{\infty}$-hypersurfaces in $\mathbb H^n$ with $n\geq 2$ are countably $\mathbb{H}^{n-1}\times\mathbb R$-rectifiabile according to Pauls' definition, even with bi-Lipschitz Maps.Comment: 30 page