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On almost-Riemannian surfaces

Roberta Ghezzi (2010/2011)

Séminaire de théorie spectrale et géométrie

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its...

On an infinite dimensional linear-quadratic problem with fixed endpoints: the continuity question

K. Maciej Przyłuski (2014)

International Journal of Applied Mathematics and Computer Science

In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study the continuity of minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

On an optimal control problem for a quasilinear parabolic equation

S. Farag, M. Farag (2000)

Applicationes Mathematicae

An optimal control problem governed by a quasilinear parabolic equation with additional constraints is investigated. The optimal control problem is converted to an optimization problem which is solved using a penalty function technique. The existence and uniqueness theorems are investigated. The derivation of formulae for the gradient of the modified function is explainedby solving the adjoint problem.

On an optimal shape design problem in conduction

José Carlos Bellido (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.

On an optimization problem arising from probability density estimation.

Sankar Basu, Mohammad Saif Ullah Khan, C.A. Micchelli, Peder A. Olsen (2002)

RACSAM

Consideramos una clase de problemas de optimización que surgen en estimaciones de la densidad de datos en dimensión elevada a partir de proyecciones en subespacios de dimensión más baja. Los criterios que se usan para la selección óptima del modelo son máxima entropía y máxima verosimilitud. En cada caso nuestro planteamiento requiere estimadores de la densidad univariados y a este respecto exploramos el uso de modelos mezcla de densidades gaussianas y de estimadores de Parzen para los datos proyectados....

On asymptotic exit-time control problems lacking coercivity

M. Motta, C. Sartori (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem....

On closure of the pre-images of families of mappings

Oleg Zaytsev (1998)

Commentationes Mathematicae Universitatis Carolinae

The closures of the pre-images associated with families of mappings in different topologies of normed spaces are considered. The question of finding a description of these closures by means of families of the same kind as original ones is studied. It is shown that for the case of the weak topology this question may be reduced to finding an appropriate closure of a given family. There are discussed various situations when the description may be obtained for the case of the strong topology. An example...

On complexity and motion planning for co-rank one sub-riemannian metrics

Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the motion planning problem for generic sub-riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [10, 11]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C ∞ case, we study some non-generic generalizations in the analytic case.

On complexity and motion planning for co-rank one sub-Riemannian metrics

Cutberto Romero-Meléndez, Jean Paul Gauthier, Felipe Monroy-Pérez (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the motion planning problem for generic sub-Riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [CITE]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C∞ case, we study some non-generic generalizations in the analytic case.

On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz, Soheil Malekzadeh (2015)

Analysis and Geometry in Metric Spaces

We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

On control problems of minimum time for Lagrangian systems similar to a swing. I. Convexity criteria for sets

Aldo Bressan, Monica Motta (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

One establishes some convexity criteria for sets in R 2 . They will be applied in a further Note to treat the existence of solutions to minimum time problems for certain Lagrangian systems referred to two coordinates, one of which is used as a control. These problems regard the swing or the ski.

On control problems of minimum time for Lagrangian systems similar to a swing. II Application of convexity criteria to certain minimum time problems

Aldo Bressan, Monica Motta (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

This Note is the Part II of a previous Note with the same title. One refers to holonomic systems Σ = A ⁢ ⋃ U with two degrees of freedom, where the part A can schemetize a swing or a pair of skis and U schemetizes whom uses A . The behaviour of U is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on Σ minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition Γ which implies...

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