General energy decay rates for a weakly damped Timoshenko system

In this paper, we consider the following Timoshenko-type system:

Exact Controllability of a Hybrid System, Membrane with Strings, on General Polygon Domains

We use here HUM (cf. Lions [9]–[l0]) to study the Neumann controllability of a two-dimensional hybrid system membrane with strings on general convex polygon domains (cf. Lee and You [1], Littman [11] for a related version of this model). This system is governed by u _tt – u = 0 in on on ₂ × (0,T), u = 0 on ₃ × (0,T); u(A _j ) = 0 if if e _{j 2} and e _{j+1 1}, 0<t<T, and if e _{j 1} and e _{j+1 2}, 0<t<T (see Sec. 1 for notations). An inverse inequality of the energy has been derived when satisfies certain geometric conditions and T is sufficiently large. As a consequence, an exact control in or is respectively obtained. Some other interesting properties (such as the uniqueness of the solution and a Carleman type inequality) of the above problems are also presented.     

Foliation Admitting Recurrent Leaves of Infinite Depth on Compact Two-Manifolds

Let be a foliation on a two-manifold M. Denote the topology closure of each leaf L of by . A sequence of proper inclusions , where each L _i is a recurrent leaf of , is called a nest of length k. The maximal length of various nests is known as the depth of the foliations . It is well known that if is orientable and M is compact, the depth of is at most one. In this paper, we show that on any orientable, compact two-manifold, there exist nonorientable foliations of infinite depth. This work negatively answers the Aranson conjecture [1]. This work was partially supported by FAPESP-Proj. Tematico No. 03/03107-9.     

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.     

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

Controllability of Systems on Solvable Lie Groups: The Generic Case

A Lie group G with Lie algebra is called SID-controllable if there exist such that the (Single Input with Drift) control system , is controllable. This is equivalent to saying that the semigroup generated by is all of G. This definition is due to Sachkov who also classified SID-controllable solvable Lie algebras, cf. [7]-[9], [11]. It turns out that SID-controllability is actually a property of the Lie algebra (rather than of a control system): if a solvable is SID-controllable, then a generic SID-system will be controllable. In this paper we generalize this result to systems with multiple inputs and drifts: G is I _n D _m-controllable if there exist inputs and drifts such that

The Deligne–Simpson Problem and the Connectedness of some Varieties

The Deligne–Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes or so that there exist irreducible (p+1)-tuples of matrices M _j ∈ C _j or A _j ∈ c _j satisfying the equality M ₁ ⋯ M _p+1 = I or A ₁ + ⋯ + A _p+1 = 0. The matrices M _j and A _j are interpreted as monodromy operators of regular linear systems and as matrices-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset or of the variety or consisting of all irreducible (p+1)-tuples (if nonempty) is connected. “Simple” means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M _j or A _j , and with a given eigenvalue is 1. To the memory of my mother     

Sphere Bundles Transverse to Holomorphic Vector Fields

We prove that for every pair of nonzero complex numbers λ ₁ and λ ₂ with \(\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}\) there is an embedding \(S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}\) transverse to the linear holomorphic vector field \(Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}\) . This extends a previous result by Ito (1989).     

Analytic Solutions of Systems

In this paper, functional series solutions of the nonlinear analytic system for the unknown state variable x(t), and functional series solutions of the analytic infinite-dimension

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.     

Global Gradient Estimates for Nonlinear Elliptic Equations with Vanishing Neumann Data in a Convex Domain

In this paper, we obtain the following global L ^q estimates

$$\left|\mathbf{f}\right|^{p } \in L^{q}({\Omega}) \Rightarrow \left|\nabla u\right|^{p } \in L^{q}({\Omega}) \quad \text{for any} ~~q\ge 1 $$

in a convex domain Ω of weak solutions for nonlinear elliptic equations of p-Laplacian type with vanishing Neumann data

$$\begin{array}{@{}rcl@{}} \text{div} \left( \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \right) & =& \text{div} \left( | \mathbf{f}|^{p-2} \mathbf{f} \right) \quad\text{in} ~~{\Omega},\\ \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \cdot \mathbf{\nu} &=& | \mathbf{f}|^{p -2} \mathbf{f}\cdot \mathbf{\nu} \quad \quad \text{on}~~ \partial{\Omega}, \end{array} $$

where ν is the outwardpointing unit normal to ?Ω. Our argument is based on the works of Banerjee and Lewis (Nonlinear Anal 100:78–85, 2014), Kinnunen and Zhou (Comm Partial Differential Equations 24(11&12):2043–2068, 1999, Differential and Integral Equations 14(4):475–492, 2001), and Byun, Wang, and Zhou (Comm Pure Appl Math 57(10):1283–1310, 2004, J Funct Anal 20(3):617–637, 2007). In the proof of the above result, we only focus on the boundary case while the interior case can be obtained as a corollary.

     

Lower and Upper Bounds for the Blow-Up Time of a Class of Damped Fourth-Order Nonlinear Evolution Equations

This paper is concerned with the study of the nonlinear damped wave equation

$$u_{tt}+{\Delta}^{2}u-{\Delta} u-\omega{\Delta} u_{t}+\alpha(t)u_{t}=\left\vert u\right\vert^{p-2}u, $$

in a bounded domain with smooth boundary. The blow-up of solutions are investigated under some conditions. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.

     

Optimal Control on the Heisenberg Group

Let H denote either the Heisenberg group , or the Cartesian product of n copies of the three-dimensional Heisenberg group . Let {X ₁, Y ₁, ...;, X _n, Y _n} be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics the cost functional with arbitrary positive parameters ₁, ...;, _n , and admissible controls taken from the set of measurable functions The above control system is encoded either in the kernel of a contact 1-form (for ), or in the kernel of a Pfaffian system (for ). In both cases, the action of the semi-direct product of the torus T ⁿ with H describe the symmetries of the problem.The Pontryagin maximum principle provides optimal controls; extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant.An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.     

On Topological Equivalence of Linear Lows with Applications to Bilinear Control Systems

This paper classifies continuous linear flows using concepts and techniques from topological dynamics. Specifically, the concepts of equivalence and conjugacy are adapted to flows on vector bundles, and the Lyapunov decomposition is characterized using the induced flows on the Grassmann and the flag bundles. These results are then applied to bilinear control systems, for which their behavior in , on the projective space , and on the Grassmannians is characterized. This research was partially supported by Proyecto FONDECYT No. 1060981 and Proyecto FONDECYT de Incentivo a la Cooperación Internacional No. 7020439.     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 0 is continuous in ? ^N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.     

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.     

Rolling of manifolds without spinning

The control model of rolling of a Riemannian manifold (M; g) onto another one $ \left( {\hat{M},\hat{g}} \right) $ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution $ {\mathcal D} $ _R, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution $ {\mathcal D} $ _R can be built as a sub-distribution of a so-called no-spinning distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to $ {{\mathcal{D}}_{\overline{\nabla}}} $ and, in particular, the geometry of $ {{\mathcal{D}}_{\overline{\nabla}}} $ -orbits. Moreover, the definition of $ {{\mathcal{D}}_{\overline{\nabla}}} $ (contrary to the definition of $ {\mathcal D} $ _R) makes sense in the general context of vector bundles equipped with linear connections. The purpose of this paper is to study the distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ determined by the product connection $ \nabla \times \hat{\nabla} $ on a tensor bundle $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ induced by linear connections ?, $ \hat{\nabla} $ on vector bundles $ E\to M,\,\,\,\hat{E}\to \hat{M} $ . We describe completely the orbit structure of $ {{\mathcal{D}}_{\overline{\nabla}}} $ in terms of the holonomy groups of ?, $ \hat{\nabla} $ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in $ {E^{*}}\otimes \hat{E} $ in terms of $ {{\mathcal{D}}_{\overline{\nabla}}} $ and the vertical tangent distribution of $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ . In the particular case of tangent bundles $ TM\to M,\,\,\,T\hat{M}\to \hat{M} $ and Levi-Civita connections, we describe in more detail how $ {{\mathcal{D}}_{\overline{\nabla}}} $ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.     

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.