Showing papers on "Uniform boundedness published in 2013"

On Event Triggered Tracking for Nonlinear Systems

Abstract: In this technical note, we study an event-based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event-based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the technical note are illustrated through simulations of trajectory tracking by a nonlinear system.

On Event Triggered Tracking for Nonlinear Systems

Abstract: In this paper we study an event based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the paper are illustrated through simulations of trajectory tracking by a nonlinear system.

Reconstruction From Anisotropic Random Measurements

Abstract: Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. This paper discusses the recently introduced restricted eigenvalue (RE) condition, which is among the most general assumptions on the matrix, guaranteeing recovery. We prove a reduction principle showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces. This principle allows us to establish the RE condition for several broad classes of random matrices with dependent entries, including random matrices with sub-Gaussian rows and nontrivial covariance structure, as well as matrices with independent rows, and uniformly bounded entries.

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

Abstract: Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions $CD(K,\infty)$ and $RCD(K,\infty)$ are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the $L^2$-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted $\Gamma$-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the $RCD(K,\infty)$ condition with $K>0$. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

Exact thresholds for Ising-Gibbs samplers on general graphs

Abstract: We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if (d−1) tanh β < 1, then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by β, and arbitrary external fields are bounded by Cnlogn. Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when dtanh β < 1, with high probability over the Erdős–Renyi random graph G(n,d/n), it holds that the mixing time of Gibbs samplers is n1+Θ(1/loglogn). Both results are tight, as it is known that the mixing time for random regular and Erdős–Renyi random graphs is, with high probability, exponential in n when (d−1) tanh β> 1, and dtanh β>1, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.

Uniform hyperbolicity of the graphs of curves

Abstract: Let C.Sg;p/ denote the curve complex of the closed orientable surface of genus g with p punctures. Masur and Minksy and subsequently Bowditch showed that C.Sg;p/ is i–hyperbolic for some i D i.g;p/ . In this paper, we show that there exists some i > 0 independent of g;p such that the curve graph C1.Sg;p/ is i– hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with g and p : the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichmuller space to C.S/ sending a Riemann surface to the curve(s) of shortest extremal length.

Covariance estimation for distributions with 2+ε moments

Abstract: We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound N=O(n) for every distribution whose k-dimensional marginals have uniformly bounded 2+e moments outside the sphere of radius O(k√). In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan–Lovasz–Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption—uniformly bounded 2+e moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704–1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.

Bounded regret in stochastic multi-armed bandits

Abstract: We study the stochastic multi-armed bandit problem when one knows the value µ (⋆) of an optimal arm, as a well as a positive lower bound on the smallest positive gap�. We propose a new randomized policy that attains a regret uniformly bounded over timein this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows �, and bounded regret of order 1/� is not possible if one only knows µ (⋆) .

Prior-free and prior-dependent regret bounds for Thompson Sampling

Abstract: We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit as the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. Building on the techniques of Audibert and Bubeck [2009] and Russo and Roy [2013] we first show that Thompson Sampling attains an optimal prior-free bound in the sense that for any prior distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This result is unimprovable in the sense that there exists a prior distribution such that any algorithm has a Bayesian regret bounded from below by $\frac{1}{20} \sqrt{n K}$. We also study the case of priors for the setting of Bubeck et al. [2013] (where the optimal mean is known as well as a lower bound on the smallest gap) and we show that in this case the regret of Thompson Sampling is in fact uniformly bounded over time, thus showing that Thompson Sampling can greatly take advantage of the nice properties of these priors.

Covering Numbers for Convex Functions

Abstract: In this paper, we study the covering numbers of the space of convex and uniformly bounded functions in multidimension. We find optimal upper and lower bounds for the $\epsilon $ -covering number of $ {\cal C}([a, b]^{d}, B)$ , in the $L_{p}$ -metric, $1 \leq p , in terms of the relevant constants, where $d \geq 1$ , $a , $B > 0$ , and $ {\cal C}([a,b]^{d}, B)$ denotes the set of all convex functions on $[a, b]^{d}$ that are uniformly bounded by $B$ . We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.

Direct adaptive neural tracking control for a class of stochastic pure‐feedback nonlinear systems with unknown dead‐zone

Abstract: SUMMARY This paper considers the problem of adaptive neural tracking control for a class of nonlinear stochastic pure-feedback systems with unknown dead zone. Based on the radial basis function neural networks' online approximation capability, a novel adaptive neural controller is presented via backstepping technique. It is shown that the proposed controller guarantees that all the signals of the closed-loop system are semi-globally, uniformly bounded in probability, and the tracking error converges to an arbitrarily small neighborhood around the origin in the sense of mean quartic value. Simulation results further illustrate the effectiveness of the suggested control scheme. Copyright © 2012 John Wiley & Sons, Ltd.

Robust adaptive asymptotic tracking control of uncertain nonlinear systems subject to nonsmooth actuator nonlinearities

Abstract: SUMMARY In this paper, we control a class of uncertain nonlinear systems preceded by nonsmooth actuator nonlinearities such as dead zone, backlash, or hysteresis. Such nonlinearities will be handled in a unified framework by using a linear model with time-varying parameters to approximate their input–output characteristics. Two robust adaptive control schemes based on Nussbaum gain technique are proposed. The controllers can guarantee global uniform boundedness of all signals and also asymptotic tracking. Simulation results also illustrate the effectiveness of the proposed schemes. Copyright © 2012 John Wiley & Sons, Ltd.

Geometry and dynamics of admissible metrics in measure spaces

Abstract: We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian

Abstract: In this article, we consider a nonlocal problem involving the p(x)-Laplacian of the type Applying a three critical points theorem from Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), pp. 249–258], we obtain the existence of two intervals of positive real parameters λ for which the above problem admits three weak solutions in , whose norms are uniformly bounded with respect to λ belonging to one of the two open intervals. In particular, we also prove some properties of p(x)-Kirchhoff–Laplace operator.

Nonlinear parabolic problems in Musielak--Orlicz spaces

Abstract: Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent $N$--function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic H\"older regularity with respect to $t$ and $x$.

Empirical Processes, Typical Sequences, and Coordinated Actions in Standard Borel Spaces

Abstract: This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable “test functions.” In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers) and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios, in which the relevant operational criteria pertain to reproducing empirical averages of a general-alphabet stationary memoryless source with respect to a suitable function class.

Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order

Abstract: In this paper, a complete Lyapunov functional was con- structed and used to obtain criteria (when p = 0) for uniform asymptotic stability of the zero solution of the nonlinear delay differential equation (1.1). When p ≠ 0, sufficient conditions are also established for uni- form boundedness and uniform ultimate boundedness of solutions of this equation. Our results improve and extend some well known results in the literature.

Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

Abstract: This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.

Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods

Abstract: This short communication analyses a boundedness property of the inverse of a Jacobian matrix that arises in regularized primal-dual interior-point methods for linear and nonlinear programming. This result should be a useful tool for the convergence analysis of these kinds of methods.

Discrete admissibility and exponential trichotomy of dynamical systems

Abstract: The aim of this paper is to present a new and complex study on the asymptotic behavior of dynamical systems, providing necessary and sufficient conditions for the existence of the exponential trichotomy. We associate to a nonautonomous discrete dynamical system an input-output system and we define a new admissibility concept called $(l^\infty(\mathbb{Z}, X), l^1(\mathbb{Z}, X))$-admissibility. First, we prove that the admissibility is a sufficient condition for the existence of the trichotomy projections, for their uniform boundedness, for their compatibility with the coefficients of the initial dynamical system and for certain reversibility properties. Assuming that the associated input-output operators satisfy a natural boundedness condition, we deduce that the admissibility is a necessary and sufficient condition for the existence of the uniform exponential trichotomy. Next, based on admissibility arguments, we obtain, for the first time in the literature, that all the trichotomic properties of a nonautonomous system can be completely recovered from the trichotomic behavior of the associated discrete dynamical system. Finally, we apply the main results in order to obtain a new characterization for uniform exponential trichotomy of evolution families in terms of discrete admissibility.

Nonconvex Optimal Control and Variational Problems

Abstract: Preface.- 1. Introduction.- 2. Well-posedness of Optimal Control Problems.- 3. Well-posedness and Porosity.- 4. Well-posedness of Nonconvex Variational Problems.- 5. Gerenic Well-posedness result.- 6. Nonoccurrence of the Lavrentiev Phenomenon for Variational Problems.- 7. Nonoccurrence of the Lavrentiev Phenomenon in Optimal Control.- 8. Generic Nonoccurrence of the Lavrentiev phenomenon.- 9. Infinite Dimensional Linear Control Problems.- 10. Uniform Boundedness of Approximate Solutions of Variational Problems.- 11. The Turnpike Property for Approximate Solutions.- 12. A Turnpike Result For Optimal Control Systems.- References.- Index.

Global behavior of solutions in a Lotka–Volterra predator–prey model with prey-stage structure

Abstract: In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large.

Strong convergence of two-dimensional Walsh-Fourier series

Abstract: We prove that certain means of quadratic partial sums of the two-dimensional Walsh–Fourier series are uniformly bounded operators acting from the Hardy space H p to the space L p for 0 < p < 1.

Regularizations of general singular integral operators.

Abstract: In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in L^1) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider regularized kernels, where the cut-off function is zero in a neighborhood of the origin, so the corresponding regularized operators with kernel are well defined (at least on a dense set). Then one can ask about uniform boundedness of the regularized operators. For the standard regularizations one usually considers truncated operators. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderon-Zygmund operators in non-homogeneous two weight settings), the L^p boundedness of the bilinear form on the compactly supported functions with separated supports (the so-called restricted L^p boundedness) implies the uniform L^p-boundedness of regularized operators for any reasonable choice of a smooth cut-off of the kernel. If the kernel satisfies some additional assumptions (which are satisfied for classical singular integral operators like Hilbert Transform, Cauchy Transform, Ahlfors--Beurling Transform, Generalized Riesz Transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators.