Bounded Functions Research Articles

ОБ ОБОБЩЕНИЯХ И ПРИЛОЖЕНИЯХ ВАРИАЦИОННЫХ ПРИНЦИПОВ НЕЛИНЕЙНОГО АНАЛИЗА

There are considered some classes of functions to which variational principles of nonlinear are applicable. In particular, it is shown that the Bishop-Phelps variational principle is applicable to some unbounded below functions. The properties of locally Lipschitzian mappings are investigated. Conditions for a mapping that is pseudo-Lipschitzian at every point of its graph to be Lipschitzian are derived.

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

ABSTRACTThis paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.

A nonparametric maximum likelihood approach for survival data with observed cured subjects, left truncation and right-censoring.

We consider observational studies in pregnancy where the outcome of interest is spontaneous abortion (SAB). This at first sight is a binary 'yes' or 'no' variable, albeit there is left truncation as well as right-censoring in the data. Women who do not experience SAB by gestational week 20 are 'cured' from SAB by definition, that is, they are no longer at risk. Our data is different from the common cure data in the literature, where the cured subjects are always right-censored and not actually observed to be cured. We consider a commonly used cure rate model, with the likelihood function tailored specifically to our data. We develop a conditional nonparametric maximum likelihood approach. To tackle the computational challenge we adopt an EM algorithm making use of "ghost copies" of the data, and a closed form variance estimator is derived. Under suitable assumptions, we prove the consistency of the resulting estimator which involves an unbounded cumulative baseline hazard function, as well as the asymptotic normality. Simulation results are carried out to evaluate the finite sample performance. We present the analysis of the motivating SAB study to illustrate the advantages of our model addressing both occurrence and timing of SAB, as compared to existing approaches in practice.

Persistently unbounded functions on locally compact Abelian groups

AbstractThis paper deals, given an arbitrary locally compact Abelian group, with the existence of integrable functions all whose convolution powers are essentially unbounded.

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

A DSC method for strict-feedback nonlinear systems with possibly unbounded control gain functions

In dynamic surface control (DSC) methods, the control gain functions of systems are always assumed to be bounded, which is a restrictive assumption. This work proposes a novel DSC approach for an extended class of strict-feedback nonlinear systems whose control gain functions are continuous and possibly unbounded. Appropriate compact sets are constructed in such a way that the trajectories of the closed-loop system do not leave these sets, therefore, in these sets, maximums and minimums values of the continuous control gain functions are well defined even if the control gain functions are possibly unbounded. By using Lyapunov theory and invariant set theory, semi-globally uniformly ultimately boundedness is analytically proved: all the signals of closed-loop system will always stay in these compact sets, while the tracking error is shown to converge to a residual set that can be made as small as desired by adjusting design parameters appropriately. Finally, the effectiveness of the designed method is demonstrated via two examples.

Approximation of Unbounded Functions by Trigonometric Polynomials

Approximation of Unbounded Functions by Trigonometric Polynomials

Robust Adaptive State Constraint Control for Uncertain Switched High-Order Nonlinear Systems

Under the weaker conditions on system power and nonlinear functions, this paper focuses on the robust adaptive state constraint control for a class of uncertain switched high-order nonlinear systems. With the help of $p$ -times differentiable unbounded function, a robust adaptive state-feedback controller is designed by combining the homogeneous domination approach with parameter separation principle. When the initial condition satisfies a suitable requirement, it is shown that all signals of the closed-loop system are bounded and system state is within a prespecified limit range by the proposed controller. As a practical application, the design scheme is utilized to the continuous stirred tank reactor with two modes feed stream. To further demonstrate the efficiency of the proposed controller, another numerical example is given.

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

Describing finite groups by short first-order sentences

We say that a class of finite structures for a finite first-order signature is r-compressible for an unbounded function r: N → N+ if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log3-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is log3-compressible. The results rely on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology. We also indicate why the results are close to optimal.

Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic

For a local field K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H-orbit in SL(d, K)/SL(d, Z) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss. We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class Cα.

Uniform continuity and quantization on bounded symmetric domains

We consider Toeplitz operators $T_f^{\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset \mathbb{C}^n$. Here $\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{\|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $\Omega=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(\Omega)$ of all $\beta$-uniformly continuous functions on $\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\Omega=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\Omega$.

On the convergence of optimal actions for Markov decision processes and the optimality of (s, S) inventory policies

AbstractThis article studies convergence properties of optimal values and actions for discounted and average‐cost Markov decision processes (MDPs) with weakly continuous transition probabilities and applies these properties to the stochastic periodic‐review inventory control problem with backorders, positive setup costs, and convex holding/backordering costs. The following results are established for MDPs with possibly non‐compact action sets and unbounded cost functions: (i) convergence of value iterations to optimal values for discounted problems with possibly non‐zero terminal costs, (ii) convergence of optimal finite‐horizon actions to optimal infinite‐horizon actions for total discounted costs, as the time horizon tends to infinity, and (iii) convergence of optimal discount‐cost actions to optimal average‐cost actions for infinite‐horizon problems, as the discount factor tends to 1. Being applied to the setup‐cost inventory control problem, the general results on MDPs imply the optimality of (s, S) policies and convergence properties of optimal thresholds. In particular this article analyzes the setup‐cost inventory control problem without two assumptions often used in the literature: (a) the demand is either discrete or continuous or (b) the backordering cost is higher than the cost of backordered inventory if the amount of backordered inventory is large.© 2017 Wiley Periodicals, Inc. Naval Research Logistics 65: 619–637, 2018

Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary $\alpha$-dependent sequences

We study the Wasserstein distance of order 1 between the empirical distribution and the marginal distribution of stationary ↵-dependent sequences. We prove some moments inequalities of order p for any p 1, and we give some conditions under which the central limit theorem holds. We apply our results to unbounded functions of expanding maps of the interval with a neutral fixed point at zero. The moment inequalities for the Wasserstein distance are similar to the well known von Bahr-Esseen or Rosenthal bounds for partial sums, and seem to be new even in the case of independent and identically distributed random variables.

A note on Markov perfect equilibria in a class of non-stationary stochastic bequest games

In this note, we prove the existence of a Markov perfect equilibrium in a non-stationary version of a paternalistic bequest game. The method we advocate is general and allows to study models with unbounded state space and unbounded utility functions. We cover both, the stochastic and deterministic cases. We provide a characterization of the set of all Markov perfect equilibria by means of a set-valued recursive equation involving the best response operator. In the stationary case, we show that there exists a set of strategies that is invariant under the best response mapping.

On approximation and implementation of transformation based feedback laws for distributed parameter systems

This paper presents an approach for the efficient numerical implementation of transformation based state feedback laws for linear distributed parameter systems. The control laws considered may originate from backstepping or flatness-based design methods. They are, therefore, directly based on the underlying distributed parameter systems. In general, they are given as unbounded functionals on the infinite-dimensional state-space. By approximating a carefully chosen bounded part, the implementation of these feedback-operators can be considerably simplified. This is achieved by approximating the state in appropriate finite-dimensional sub-spaces of the state-space. The choice of these sub-spaces as well as the controller implementation are discussed for both, a particular motivating example specific and the general case. For the implementation and validation of the obtained approximated controllers the python-based software toolbox PyInduct is introduced.

Adaptive robust output tracking of uncertain strict-feedback nonlinear time-delay systems via control schemes with simple structure

ABSTRACTThe output tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with time-varying delays. In the paper, the time-varying delays are assumed to be any non-negative continuous and bounded functions, and it is not necessary for their derivatives to be less than one. It is also assumed that the upper bounds of nonlinear delayed state perturbations and external disturbances are unknown. On the basis of backstepping algorithm, a novel design method is proposed by which some simple adaptive robust output tracking control schemes are synthesised. The proposed design method can avoid the repeated differentiation problem which appears in using the conventional backstepping algorithm, and need not know all the nonlinear upper bound functions of uncertainties, which are repeatedly employed at each step of the backstepping algorithm. In particular, it is not necessary to know any information on the time-varying delays to construct our simple output tracking control schemes. It is also shown that the tracking error can converge uniformly exponentially towards a neighbourhood of the origin. Finally, a numerical example and its simulations are provided to demonstrate the design procedure of the simple method proposed in the paper.

A sharp first order analysis of Feynman–Kac particle models, Part I: Propagation of chaos

This article provides a new theory for the analysis of forward and backward particle approximations of Feynman–Kac models. Such formulae are found in a wide variety of applications and their numerical (particle) approximation is required due to their intractability. Under mild assumptions, we provide sharp and non-asymptotic first order expansions of these particle methods, potentially on path space and for possibly unbounded functions. These expansions allow one to consider upper and lower bound bias type estimates for a given time horizon n and particle number N; these non-asymptotic estimates are O(n∕N). Our approach is extended to tensor products of particle density profiles, leading to new sharp and non-asymptotic propagation of chaos estimates. The resulting upper and lower bound propagations of chaos estimates seem to be the first result of this kind for mean field particle models.

Constructive proof of Lagrange stability and sufficient – Necessary conditions of Lyapunov stability for Yang–Chen chaotic system

This paper studies the stability problem of Yang–Chen system. By introducing different radial unbounded Lyapunov functions in different regions, global exponential attractive set of Yang–Chen chaotic system is constructed with geometrical and algebraic methods. Then, simple algebraic sufficient and necessary conditions of global exponential stability, global asymptotic stability, and exponential instability of equilibrium are proposed. And the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability, exponential instability of equilibria are obtained. Furthermore, the branch problem of the system is discussed, some branch expressions are given for the parameters of the system.

Stability with respect to initial conditions in V-norm for nonlinear filters with ergodic observations

AbstractWe establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in V-norm, allowing for unbounded test functions. The analysis is conducted in an general setup involving nonnegative kernels in a random environment which allows treatment of filters and prediction filters in a single framework. The main result is illustrated on two examples, the first showing that a total variation norm stability result obtained by Douc et al. (2009) can be extended to V-norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in V-norm for the filters, but the V-norm of each prediction filter is infinite.