Boundedness Results Research Articles

On Output Tracking Using Dynamic Output Feedback Discrete-Time Sliding-Mode Controllers

In this note, an output feedback based discrete-time sliding-mode control scheme is proposed. It incorporates a steady-state tracking requirement through the use of integral action. Previous work has shown that with an appropriate choice of sliding surface, discrete-time sliding-mode control can be applied to nonminimum phase systems. The original scheme employed static output feedback and this imposed restrictions on the class of systems to which it was applicable - specifically a certain "fictitious" subsystem was required to be output feedback stabilizable. The scheme proposed in this note includes a compensator which broadens the class of systems for which the results are applicable. In the presence of bounded matched disturbances, ultimate boundedness results are obtained. It is also shown that in the presence of a class of sector bounded uncertainty, asymptotic stability can be achieved.

Weak inequalities for Cesàro and Riesz summability of Walsh–Fourier series

The maximal operators for Cesàro or ( C , α ) and Riesz summability with respect to Walsh–Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from H p to L p for all 1 / ( α + 1 ) < p < ∞ . In this work we prove that this boundedness result does not hold anymore if p ⩽ 1 / ( α + 1 ) . However, for p = 1 / ( α + 1 ) the maximal operators are bounded from H 1 / ( α + 1 ) to the weak L 1 / ( α + 1 ) space. To the proof some known estimations for the Cesàro and Riesz kernels have to be sharpened.

Moduli of twisted sheaves

We study moduli of semistable twisted sheaves on smooth proper algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asymptotically geometrically irreducible, normal, generically smooth, and locally complete intersections (l.c.i.'s) over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities, semistability and boundedness results, and basic results on twisted Quot-schemes on a surface

Boundedness for commutators of fractional Hardy operators on Herz spaces

Some boundedness results are established for commutators generated by classical fractional Hardy operators and one-sided CMO functions on the homogeneous Herz spaces, which extends some known results.

Boundedness and Large-Time Behavior Results for a Diffusive Epidemic Model

We consider a reaction-diffusion system modeling the spread of an epidemic disease within a population divided into the susceptible and infective classes. We first consider the question of the uniform boundedness of the solutions for which we give a positive answer. Then we deal with the asymptotic behavior of the solutions where in particular we are interested in reasonable conditions leading to the extinction of the infection disease as the time goes to infinity.

Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H dim(X)−2 and H dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.

Weighted Weak-type Estimates for Multilinear Commutators of Fractional Integrals on Spaces of Homogeneous Type

We obtain weighted distributional inequalities for multilinear commutators of the fractional integral on spaces of homogeneous type. The techniques developed in this work involve the behavior of some fractional maximal functions. In relation to these operators, as a main tool, we prove a weighted weak type boundedness result, which is interesting in itself.

Local estimates on two linear parabolic equations with singular coefficients

We treat the heat equation with singular drift terms and its generalization: the linearized Navier-Stokes system. In the first case, we obtain boundedness of weak solutions for highly singular, supercritical data. In the second case, we obtain regularity results for weak solutions with mildly singular data (those in the Kato class). This not only extends some of the classical regularity theory from the case of elliptic and heat equations to that of linearized Navier-Stokes equations but also proves an unexpected gradient estimate, which extends the recent interesting boundedness result of O'Leary.

On one-sided potentials with multiple kernels

Necessary and sufficient conditions for weight function v governing the boundedness of the Volterra, truncated and ball fractional integral operators with multiple kernels from L p to are established, where 1 < p ≤ q < ∞. The result for the operator with multiple positive kernels enables us to derive a trace inequality for the Riemann–Liouville transform with α, β > 1/p. An example of weight function v presented in the work shows that the condition which arises applying twice one-dimensional boundedness result, for example, to the operator is different from the criterion guaranteeing the inequality To prove the main statements, we use higher dimensional Hardy's inequalities which are also studied in this article.

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.

Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term

We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following(1){ut−div(α(u)∇u)=β(u)|∇u|2+f(x,t),in Ω×]0,T[;u(x,t)=0,on ∂Ω×]0,T[;u(x,0)=u0(x),in Ω. Here Ω is a bounded open set in RN, T>0. The unknown function u=u(x,t) depends on x∈Ω and t∈]0,T[. The symbol ∇u denotes the gradient of u with respect to x. The real functions α, β are continuous; moreover α is positive, bounded and may vanish at ±∞. As far as the data are concerned, we require the following assumptions:∫ΩΦ(u0(x))dx<∞ where Φ is a convenient function which is superlinear at ±∞ andf(x,t)∈Lr(0,T;Lq(Ω))with 1r+N2q⩽1. We give sufficient conditions on α and β in order to have distributional solutions. We point out that the assumptions on the data do not guarantee in general the boundedness of the solutions; this means that the coercivity of the principal part of the operator can really degenerate. Moreover, a boundedness result is proved when the assumptions on the data are strengthened.

Maximal operators and Fourier transforms of self-similar measures

Abstract A self-similar measure on R n is defined to be a probability measure satisfying μ = ∑ j = 1 N p j μ ∘ S j - 1 + ∑ j = 1 M q j ( μ ∗ μ ) ∘ T j - 1 , where Sjx = ρjRjx + bj, Tjx = ηjQjx + cj are contractive similarities, 0 ρ j 1 , 0 η j 1 2 , 0 p j 1 , 0 q j 1 , ∑ j = 1 N p j + ∑ j = 1 M q j = 1 , Rj, Qj are orthogonal matrix and μ ∗ μ is the convolution of two measures. When M = 0, μ is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of μ, such as ∫ | x | ⩽ R ∂ ∂ x α μ ˆ ( x ) 2 d x = O ( R n - β ) for any order derivation of μ ˆ ( x ) as R → ∞ under certain additional hypotheses. When M > 0, μ is a nonlinear self-similar measure, we get some results of Lp boundedness for maximal operators of μ, from the pointwise asymptotic estimate of the Fourier transform of μ made by Strichartz.

Boundedness of rough singular integral operatorson the homogeneous Morrey-Herz spaces

Some boundedness results are established for some rough singular integral operators and fractional singular integral operators on the Homogeneous Morrey-Herz Spaces. And also we get some boundedness results on the weak homogeneous Morrey-Herz Spaces.

BOUNDS ON COINCIDENCE INDICES ON NON-ORIENTABLE SURFACES

This paper presents some results about bounds for coincidence indices of Nielsen coincidence classes for maps between nonorientable surfaces. Denoting by Kn the nonorientable surface constructed by a connected sum of n torus with a Klein bottle, the author proves: (i) for pairs of maps between two Klein bottles or for pairs of maps from a Klein bottle to a surface Kn the coincidence class index is bounded. (ii) for pairs of maps from Kn to the Klein bottle the coincidence class index is unbounded. Other boundedness results are given for more technical conditions, including one for self maps.

MASTER-SLAVE SYNCHRONIZATION OF ROBOT MANIPULATORS: EXPERIMENTAL RESULTS

This paper presents experimental results for master-slave synchronization of two robot manipulators using a recently developed observer-controller scheme. The paper aims to investigate the value and the limitations of the theory. In particular, the theoretical result of uniform ultimate boundedness of the synchronization errors is investigated. Back-to-back simulation results are also presented.

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.

A note on the stability and boundedness results of solutions of certain fourth order differential equations

The purpose of this paper is to investigate the asymptotic stability of the zero solution of x (4)+ϕ( x ̈ ) x ⃛ +f(x, x ̇ ) x ̈ +g( x ̇ )+h(x)=0 with p≡0 and the boundedness of all solutions of x (4)+ϕ( x ̈ ) x ⃛ +f(x, x ̇ ) x ̈ +g( x ̇ )+h(x)=0 with p≠0. The results obtained here revise the results in [Demonstratio Math. 31(1) (1998) 87] and improve and includes some results in the relevant literature [J. Math. Anal. Appl. 5 (1962) 136; J. London Math. Soc. 37 (1962) 28; J. London Math. Soc. 42 (1967) 51; SIAM. J. Math. Anal. 1 (1970) 27; Int. J. Control. 69(2) (1998) 353].

Boundedness for the commutator of convolution operator

A boundedness result is established for sublinear operators on homogeneous Herz spaces. As applications, a new result about the weighted boundedness of commutators of convolution operators is obtained.

A boundedness result for twisted convolution

AbstractWe consider twisted convolution operators with kernels having singularities on a sphere and having as Fourier transform the oscillatory symbol mα(|ξ|) = |ξ|–αei|ξ|, 0 ≤ ℜ︁α &lt; 2n. We give integral representations for such operators and, as a principal result, we study Lp–Lq estimates for them.