Gauss-Bonnet Theorem in Sub-Riemannian Heisenberg Space $\mathbb H^{1}$

We prove a version of the Gauss-Bonnet theorem in sub-Riemannian Heisenberg space \(\mathbb H^{1}\). The sub-Riemannian distance makes \(\mathbb H^{1}\) a metric space that consequently has a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and, for compact surfaces (which are topologically a torus), we obtain \({\int }_{S}K=0\).     

Attractor for the nonlinear Schr?dinger equation with a nonlocal nonlinear term

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrödinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor \( \mathcal{A} \) for the nonlocal equation in the strong topology of H ¹(Ω). We also prove that the global attractor is regular, i.e., \( \mathcal{A} \subset {H^2}\left( \Omega \right) \), assuming that f(x) is of class C ². Furthermore, we estimate the number of the determining modes for this equation.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

Horizontal Holonomy for Affine Manifolds

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection ? with holonomy group H ^? and Δ a smooth completely non integrable distribution. We define the Δ-horizontal holonomy group \({H^{\;\nabla }_{\Delta }}\) as the subgroup of H ^? obtained by ?-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of \({H^{\;\nabla }_{\Delta }}\) and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that \({H^{\;\nabla }_{\Delta }}\) is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m ≥ 2 generators, and ? is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that \({H^{\;\nabla }_{\Delta }}\) is compact and strictly included in H ^? as soon as m ≥ 3.     

Genericity for Non-Wandering Surface Flows

Consider the set \(\chi ^{0}_{\text {nw}}\) of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M?Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in \(\chi ^{0}_{\text {nw}}\); (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in \(\chi ^{0}_{\text {nw}}\) if and only if the surface is homeomorphic to either the sphere \(\mathbb {S}^{2}\), the projective plane \(\mathbb {P}^{2}\), or the Klein bottle \(\mathbb {K}^{2}\).     

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.

     

On Symmetries in Time Optimal Control,Sub-Riemannian Geometries,and the <Emphasis Type="Italic">K</Emphasis>−<Emphasis Type="Italic">P</Emphasis> Problem

The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit space M/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K?P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on S O(3).     

Conformal Equivalence of 3D Contact Structures on Lie Groups

In this paper, a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular, we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group \(\mathbb {H}_{3}\) or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to S U (2,1).     

Small Time Asymptotics on the Diagonal for Hörmander’s Type Hypoelliptic Operators

We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x ₀. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x ₀, then so is also the original control problem, and in this case we show that the fundamental solution blows up as \(\phantom {\dot {i}\!}t^{-\mathcal {N}/2}\), where \(\phantom {\dot {i}\!}\mathcal {N}\) is a number determined by the Lie algebra at x ₀ of the fields, that define the hypoelliptic operator.     

Harmonic Analysis of Finite Lamplighter Random Walks

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular, in the case where the underlying graph is the infinite path \( \mathbb{Z} \). In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C ₂-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. If the graph has a transitive isometry group G, we also describe the spectral analysis in terms of the representation theory of the wreath product C ₂?G. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples have already been studied by Häggström and Jonasson by probabilistic methods.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Let X be a real Banach space and I a nonempty interval. Let \(K:I\rightsquigarrow X\) be a multi-function with the graph \(\mathcal {K} \). We give here a characterization for \(\mathcal {K} \) to be approximate/near weakly invariant with respect to the differential inclusion \(x^{\prime }(t)\in F(t, x(t))\) by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârj? et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârj? et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion \(x^{\prime }(t)\in F(t,x(t))\).     

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.     

Gevrey Order and Summability of Formal Series Solutions of Certain Classes of Inhomogeneous Linear Integro-Differential Equations with Variable Coefficients

In this article, we investigate Gevrey and summability properties of formal power series solutions of certain classes of inhomogeneous linear integro-differential equations with analytic coefficients in a neighborhood of \((0,0)\in \mathbb {C}^{2}\). In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.     

When a Minimal Map Is Totally Transitive on a <Emphasis Type="Italic">G</Emphasis>-Space

In this paper, we introduce the notion of G-regular periodic decomposition (GRPD) for maps on G-spaces and investigate its relation with G-transitivity. It is shown that if a pseudoequivariant, G-transitive map on a G-space has a GRPD of some length n, then its nth iterate is not G-transitive. On the other hand, if a pseudoequivariant, G-transitive map on a G-space has a non-G-transitive nth iterate, then it admits a GRPD of length p for some prime p dividing n. Using the notion of GRPD, it is obtained that a pseudoequivariant, G-minimal map is totally G-transitive on a connected G-space.     

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

We consider the problem P _{c u r v e} of minimizing \(\int \limits _{0}^{L} \sqrt {\xi ^{2} + \kappa ^{2}(s)} \, \mathrm {d}s\) for a curve x in \(\mathbb {R}^{3}\) with fixed boundary points and directions. Here, the total length L≥0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ>0 is constant. We lift problem P _{c u r v e} on \(\mathbb {R}^{3}\) to a sub-Riemannian problem P _{m e c} on SE(3)/({0}×SO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P _{c u r v e} . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.     

Adjoint Orbits of <Emphasis Type="Italic">sl</Emphasis>(2, ℝ) on Real Simple Lie Algebras and Controllability

This paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root α, there exists a Cartan decomposition of L, related to α, which characterizes some controllability properties by using the adjoint orbits of sl(2, ?). For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient.     

Cascade Linearization of Invariant Control Systems

Let Pfaffian system ω define an intrinsically nonlinear control system on manifold M that is invariant under the free, regular action of a Lie group G. The problem of identifying and constructing static feedback linearizable G-quotients of ω was solved in De Doná et al. (2016). Building on these results, the present paper proves that the trajectories of ω can often be expressed as the composition of the trajectories of a static feedback linearizable quotient control system, ω/G, on quotient manifold M/G, and those of a separate control system, γ ^G , evolving on a principal G-bundle over a jet space. Furthermore, we point out that ω may not only have a static feedback linearizable quotient, ω/G but additionally, γ ^G itself may possess a static feedback linearizable reduction as well. This enables one to express the trajectories of an intrinsically nonlinear control system as the composition of the trajectories of static feedback linearizable control systems, thereby providing a geometric criterion for the explicit integrability of intrinsically nonlinear systems. Moreover, special integrability properties arise when G is solvable. Examples are presented in which the above phenomena are explicitly demonstrated. An important aspect of the examples is that they gather evidence for the conjecture that our sufficient conditions for explicit integrability are also necessary.