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Über dieses Buch

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.

Inhaltsverzeichnis

Frontmatter

Some open problems

We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
Andrei A. Agrachev

Geometry of Maslov cycles

We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated situations, from real algebraic geometry to sub-Riemannian geometry.
Davide Barilari, Antonio Lerario

How to Run a Centipede: a Topological Perspective

In this paper we study the topology of the configuration space of a device with d legs (“centipede”) under some constraints, such as the impossibility to have more than k legs off the ground. We construct feedback controls stabilizing the system on a periodic gait and defined on a ‘maximal’ subset of the configuration space.
A centipede was happy quite!
Until a toad in fun
Said, “Pray, which leg moves after
which?”
This raised her doubts to such a pitch,
She fell exhausted in the ditch
Not knowing how to run.
Katherine Craster
Yuliy Baryshnikov, Boris Shapiro

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

We combine geometric and numerical techniques - the Hampath code - to compute conjugate and cut loci on Riemannian surfaces using three test bed examples: ellipsoids of revolution, general ellipsoids, and metrics with singularities on S2 associated to spin dynamics.
Bernard Bonnard, Olivier Cots, Lionel Jassionnesse

On the injectivity and nonfocal domains of the ellipsoid of revolution

In relation with regularity properties of the transport map in optimal transportation on Riemannian manifolds, convexity of injectivity and nonfocal domains is investigated on the ellipsoid of revolution. Building upon previous results [4, 5], both the oblate and prolate cases are addressed. Preliminary numerical estimates are given in the prolate situation.
Jean-Baptiste Caillau, Clément W. Royer

Null controllability in large time for the parabolic Grushin operator with singular potential

We investigate the null controllability property for the parabolic Grushin equation with an inverse square singular potential. Thanks to a Fourier decomposition for the solution of the equation, we can reduce the problem to the validity of a uniform observability inequality with respect to the Fourier frequency. Such an inequality is obtained by means of a suitable Carleman estimate, with an adapted spatial weight function. We thus show that null controllability holds in large time, as in the case of the Grushin operator without potential.
Piermarco Cannarsa, Roberto Guglielmi

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results– of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling problem.
Yacine Chitour, Mauricio Godoy Molina, Petri Kokkonen

Optimal stationary exploitation of size-structured population with intra-specific competition

We analyze an exploitation of size-structured population in stationary mode and prove the existence of stationary state of population for a given stationary control. The existence of an optimal control is proved and the necessary optimal condition is found.
Alexey A. Davydov, Anton S. Platov

On geometry of affine control systems with one input

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works for rank 2 distributions of maximal class in ℝn with additional structures such as affine control systems with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. In contrast to the case of an arbitrary rank 2 distribution without additional structures, in the considered cases each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n−3)-dimensional for arbitrary n ≥ 5. The moduli spaces of the most symmetric models are described as well.
Boris Doubrov, Igor Zelenko

Remarks on Lipschitz domains in Carnot groups

In this Note we present the basic features of the theory of Lipschitz maps within Carnot groups as it is developed in [8], and we prove that intrinsic Lipschitz domains in Carnot groups are uniform domains.
Bruno Franchi, Valentina Penso, Raul Serapioni

Differential-geometric and invariance properties of the equations of Maximum Principle (MP)

An invariant formulation of the Pontryagin Maximum Principle (PMP) is given. It is proved that the Pontryagin derivative P X coincides on vector fields X ∊ Vect M, (M - the configuration space of the problem), with the Lie bracket ad X , and the flow generated on the cotangent bundle T* M by the vector field P x is bundle-preserving.
Revaz V. Gamkrelidze

Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces

In this paper we present a survey of the joint program with Fabrice Baudoin originated with the paper [6], and continued with the works [79] and [10], joint with Baudoin, Michel Bonnefont and Isidro Munive.
Nicola Garofalo

Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds

This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.
Roberta Ghezzi, Frédéric Jean

The Delauney-Dubins Problem

The problem of Delauney, posed in the middle of the nineteenth century asked for curves of shortest and longest length among all space curves with a given constant curvature that connect two given tangential directions. About a hundred years later, L. Dubins, apparently unaware of the former problem, asked for a curve of minimal length that joins two fixed directions in the space of curves whose curvature is less or equal than a given constant. Dubins showed that the minimizers exist in the class of continuously differentiable curves having Lebesgue integrable second derivative and he characterized optimal solutions in the plane as the concatenations of circles of curvature ±c and straight lines with at most two switchings from one arc to another ( [7]). Remarkably,, the key equation in the problem of Delauney, obtained by Josepha Von Schwartz in mid 1930s also appears in the spacial version of the problem of Dubins.
In this paper we will show that the n -dimensional problem of Dubins (called Delauney-Dubins, for historical reasons) is essentially three dimensional on any space form (simply connected space of constant curvature). We also show that the extremal equations are completely integrable and consist of two kinds, switching and non-switching.The non-switching extremals are expressed in terms of elliptic functions obtained by solving the fundamental equation of Josepha Von Schwarz, while the projections of the switching extremals are shown to be the concatenations of arcs of circles (hyperbolas, in the hyperbolic case) and geodesics, exactly as in the two dimensional Dubins’ problem ([16]).
Velimir Jurdjevic

On Local Approximation Theorem on Equiregular Carnot-Carathéodory Spaces

We prove the Local Approximation Theorem on equiregular Carnot-Carathéodory spaces with C 1-smooth basis vector fields.
Maria Karmanova, Sergey Vodopyanov

On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle

The Jacobi curve of an extremal of an optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. For parametrized curves in Lagrange Grassmannians satisfying very general assumptions, the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, were constructed. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the “coefficients” of this normal form. In the present paper, we focus on the curvature maps for an optimal control problem of a natural mechanical system on a sub-Riemannian structure on a principle connection of a principle G-bundles with one dimensional fibers over a Riemannian manifold. We express the curvature maps in terms of the curvature tensor of the base Riemannian manifold and the curvature form and the potential.
Chengbo Li

On the Alexandrov Topology of sub-Lorentzian Manifolds

In the present work, we show that in contrast to sub-Riemannian geometry, in sub-Lorentzian geometry the manifold topology, the topology generated by an analogue of the Riemannian distance function and the Alexandrov topology based on causal relations, are not equivalent in general and may possess a variety of relations. We also show that ‘opened causal relations’ are more well-behaved in sub-Lorentzian settings.
Irina Markina, Stephan Wojtowytsch

The regularity problem for sub-Riemannian geodesics

We review some recent results on the regularity problem of sub-Riemannian length minimizing curves. We also discuss a new nontrivial example of singular extremal that is not length minimizing near a point where its derivative is only Hölder continuous. In the final section, we list some open problems.
Roberto Monti

A case study in strong optimality and structural stability of bang-singular extremals

Motivated by the well known dodgem car problem, we give sufficient conditions for strong local optimality and structural stability of a bang-singular trajectory in a minimum time problem where the dynamics is single input, affine with respect to the control and depends on a finite-dimensional parameter, the initial point is fixed and the final one is constrained to an integral line of the controlled vector field.
On the nominal problem, we assume the coercivity of a suitable second variation along the singular arc and regularity both of the bang arc and of the junction point, thus obtaining sufficient conditions for strict strong local optimality for the given bang-singular extremal trajectory. Moreover, assuming the uniqueness of the adjoint covector along the singular arc, we prove that, for any sufficiently small perturbation of the parameter, there is a bang-singular extremal trajectory which is a strict strong local optimiser for the perturbed problem.
The results are proven via the Hamiltonian approach to optimal control and by taking advantage of previous results of the authors.
Laura Poggiolini, Gianna Stefani

Approximate controllability of the viscous Burgers equation on the real line

The paper is devoted to studying the 1D viscous Burgers equation controlled by an external force. It is assumed that the initial state is essentially bounded, with no decay condition at infinity, and the control is a trigonometric polynomial of low degree with respect to the space variable. We construct explicitly a control space of dimension 11 that enables one to steer the system to any neighbourhood of a given final state in local topologies. The proof of this result is based on an adaptation of the Agrachev-Sarychev approach to the case of an unbounded domain.
Armen Shirikyan

Homogeneous affine line fields and affine lines in Lie algebras

We prove that for n = 2, 3 any local homogeneous affine line field L;T n can be described by an affine line ℓ in an n-dimensional Lie algebra g, which means that L is diffeomorphic to the affine line field in a neighborhood of the identity of the Lie group of g obtained by pushing ℓ along the flows of left-invariant vector fields. We show that this statement does not hold for n = 4, for one of several types of homogeneous line fields.
Michail Zhitomirskii
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