Bound Of Norm Research Articles

The Operator-Splitting Method for Cahn-Hilliard is Stable

We prove energy stability of a standard operator-splitting method for the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms of the numerical solution and convergence of the splitting approximation. This is the first energy stability result for the operator-splitting method for the Cahn-Hilliard equation. Our analysis can be extended to many other models.

On the topological solutions with vortices and antivortices for the Maxwell-Chern-Simons O(3) sigma model on a torus

In this paper, we consider the self-dual O(3) Maxwell-Chern-Simons-Higgs equations on a two-dimensional flat torus arising from the O(3) sigma gauge field model. Our main goal is to obtain the existence and uniqueness of topological solutions with vortices and antivortices. In order to achieve this goal, we show that the nondegeneracy of linearized operator for entire solution holds even when the symmetry is broken. Moreover, we also obtain the uniform bound of L1 norm of nonlinearities with respect to large charge of electron. We expect that these results would play an significant role to get the asymptotic behavior of all possible solutions and count the total number of solutions.

Robust adaptive sliding mode attitude control for aircraft systems based on back-stepping method

In this paper, the robust adaptive control and back-stepping method are considered for attitude control of aircraft systems. First, a non-linear aircraft model is introduced and converted to standard state space form. Then, an adaptive sliding mode controller is designed by using the back-stepping method. An adaptive law is designed to cooperate with sliding mode control (SMC) to compensate for the unknown and uncertain parts of the aircraft systems. This method does not need to know the physical parameters of the aircraft model accurately, but only uses the adaptive upper bound of the matrix model norm to ensure the robustness of the system under SMC. Next, the stability of the adaptive law and SMC is analyzed and proven theoretically. At last, the effectiveness of this method is illustrated by extensive comparative simulations. Furthermore, the disturbance term is introduced to test the anti-interference performance of the controller. The results acquired from simulation illustrate that SMC can achieve good control performance.

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Operator norm and lower bound of four-dimensional generalized Hausdorff matrices

Explicit inverse of near Toeplitz pentadiagonal matrices related to higher order difference operators

This paper analyzes the inverse of near Toeplitz pentadiagonal matrices, arising from a finite-difference approximation to the fourth-order nonlinear beam equation. Explicit non-recursive inverse matrix formulas and bounds of norms of the inverse matrix are derived for the clamped–free and clamped–clamped boundary conditions. The bound of norms is then used to construct a convergence bound for the fixed-point iteration of the form u=f(u) for solving the nonlinear equation. Numerical computations presented in this paper confirm the theoretical results.

Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

On norm compression inequalities for partitioned block tensors

When a tensor is partitioned into subtensors, some tensor norms of these subtensors form a tensor called a norm compression tensor. Norm compression inequalities for tensors focus on the relation of the norm of this compressed tensor to the norm of the original tensor. We prove that for the tensor spectral norm, the norm of the compressed tensor is an upper bound of the norm of the original tensor. This result can be extended to a general class of tensor spectral norms. We discuss various applications of norm compression inequalities for tensors. These inequalities improve many existing bounds of tensor norms in the literature, in particular tightening the general bound of the tensor spectral norm via tensor partitions. We study the extremal ratio between the spectral norm and the Frobenius norm of a tensor space, provide a general way to estimate its upper bound, and in particular, improve the current best upper bound for third order nonnegative tensors and symmetric tensors. We also propose a faster approach to estimate the spectral norm of a large tensor or matrix via sequential norm compression inequalities with theoretical and numerical evidence. For instance, the complexity of our algorithm for the matrix spectral norm is Oleft( n^{2+epsilon }right) where epsilon ranges from 0 to 1 depending on the partition and the estimate ranges correspondingly from some close upper bound to the exact spectral norm.

Exponential small-gain theorem and fault tolerant safe control of interconnected nonlinear systems

This paper considers a class of interconnected nonlinear systems where each subsystem, in the absence of coupling, is individually exponentially stable. The cyclic-small-gain theorem is significantly extended in such a way that the interconnected system is proved to be globally exponentially stable, an exponential converging upper bound of state norm is obtained which fully reveals the relations between the gains, the decay rate and the upper bound of the states. The new result is further applied to fault tolerant safe control problem of interconnected nonlinear systems. A fault recoverability condition with respect to safety is established, under which both individual and cooperative fault tolerant safe control strategies are provided under the decentralized control structure. This guarantees that the states are always within a given safe domain in the presence of faults.

Schur Complement-Based Infinity Norm Bounds for the Inverse of SDD Matrices

Based on the Schur complement, two upper bounds for the infinity norm of the inverse of strictly diagonally dominant matrices are presented. We apply these new bounds to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of B-matrices. A lower bound for the smallest singular value is also given.

Stability Analysis of Multivariable Digital Control Systems with Uncertain Timing

The ever increasing complexity of real-time control systems results in significant deviations in the timing of sensing and actuation, which may lead to degraded performance or even instability. In this paper we present a method to analyze stability under mostly-periodic timing with bounded uncertainty, a timing model typical for the implementation of controllers that were actually designed for strictly periodic execution. In contrast to existing work, we include the case of multiple sensors and actuators with individual timing uncertainty. Our approach is based on the discretization of a linear impulsive system. To avoid the curse of dimensionality, we apply a decomposition that breaks down the complex timing dependency into the effects of individual sensor-actuator pairs. Finally, we verify stability by norm bounding and a Common Quadratic Lyapunov Function. Experimental results substantiate the effectiveness of our approach for moderately complex systems.

Spectral bounding: Strictly satisfying the 1-Lipschitz property for generative adversarial networks

Imposing the 1-Lipschitz constraint is a problem of key importance in the training of Generative Adversarial Networks (GANs), which has been proved to productively improve stability of GAN training. Although some interesting alternative methods have been proposed to enforce the 1-Lipschitz property, these existing approaches (e.g., weight clipping, gradient penalty (GP), and spectral normalization (SN)) are only partially successful. In this paper, we propose a novel method, which we refer to as spectral bounding (SB) to strictly enforce the 1-Lipschitz constraint. Our method adopts very cost-effective terms of both 1-norm and ∞-norm, and yet allows us to efficiently approximate the upper bound of spectral norms. In this way, our method provide important insights to the relationship between an alternative of strictly satisfying the Lipschitz property and explainable training stability improvements of GAN. Our proposed method thus significantly enhances the stability of GAN training and the quality of generated images. Extensive experiments are conducted, showing that the proposed method outperforms GP and SN on both CIFAR-10 and ILSVRC2015 (ImagetNet) dataset in terms of the standard inception score.

Consensus via event-triggered strategy of nonlinear multi-agent systems with Markovian switching topologies

The consensus problem via event-triggered strategy of nonlinear multi-agent systems (MASs) with Markovian switching topologies is addressed in this paper. To simplify information transmission, a distributed event-triggered mechanism (ETM) is designed. By combining the Markovian switching topologies and ETM, an appropriate consensus control protocol is presented. Considering the impact of delay on systems stability, the system model is transformed by the delay system approach. By choosing a Lyapunov–Krasovskii function and applying linear matrix inequalities technique, sufficient conditions are obtained to ensure the consensus stability with H∞ norm bound of the MASs. A numerical simulation is given to confirm the effectiveness of the designed method.

Adaptive control for a class of nonlinear chaotic systems with quantized input delays

This paper proposes adaptive prediction-based control design for nonlinear chaotic systems in the presence of input quantization with actuator delay. A class of sector-bounded quantizer, namely the hysteresis quantizer, has been used to quantize the control signal. Adaptive scheme is utilized to estimate unknown parameters of the nonlinear chaotic system. A new prediction vector is introduced to compensate actuator delay for nonlinear systems with unknown parameters. The proposed controller does not require the restrictive conditions for quantized parameters in contrast to some existing control schemes for systems with input quantization. By error analysis, it is insured that the upper bound of the state vector norm is limited. Moreover, a maximum allowable upper bound for actuator time-delay is achieved and is guaranteed that for any input delay less than this value, the closed-loop system is stable. To show the efficiency of the proposed method, it is applied to brushless DC motor and Chua’s circuit with delayed hysteresis input and compared to the existing results in the literature. Simulation results show that the proposed method is able to compensate actuator time-delay with hysteresis quantized input for a class of nonlinear systems having unknown parameters. The robustness and efficiency of the suggested approach are illustrated by comparative simulation results.

A trivariate near-best blending quadratic quasi-interpolant

In this paper, we construct a new trivariate spline quasi-interpolation operator. It is expressed as blending sum of univariate and bivariate C1 quadratic spline quasi-interpolants and it is of near-best type, i.e. it has a small infinity norm and the coefficients functionals defining it are determined by minimizing an upper bound of the operator infinity norm, derived from the Bernstein-Bézier coefficients of its Lebesgue function.

Global existence and convergence rates of solutions for the 3D compressible magnetohydrodynamic equations without heat conductivity

ABSTRACT The Cauchy problem for the 3D compressible Navier–Stokes and magnetohydrodynamic equations without heat conductivity is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space only. But we don't need the bound of norm. Moreover, the – convergence rates are also obtained for the solution. This is different from the works of Duan and Ma [Global existence and convergence rates for the 3-D compressible Navier–Stokes equations without heat conductivity. Indiana Univ Math J. 2008;57:2299–2319] and Tan and Wang [On hyperbolic-dissipative systems of composite type. J Diff Eq. 2016;260:1091–1125]. Our proof is based on the benefit of the low-frequency and high-frequency decompositions, here, we just need spectral analysis of the low-frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

Robust Stability Analysis of Uncertain Linear Fractional-Order Systems With Time-Varying Uncertainty for 0 < α < 2

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

ОЦЕНКА РИСКА РАЗВИТИЯ СОМАТИЧЕСКИХ НАРУШЕНИЙ ЗДОРОВЬЯ У ДЕТЕЙ С ПОЛИГИПОВИТАМИНОЗОМ

This paper deals with study of the seasonal vitamin provision for 188 organized children attending pre-school educational institutions with standard main meal C-fortification. The study showed that 75–85 % of children have year-round vitamin deficiency, 40 % of whom have it in the nature of polyhypo-vitaminosis. The study has shown that even in the autumn-winter period the basic vitamin provision for children does not exceed the lower bound of the physiological norm, and during the spring months about 70 per cent of children suffer from subclinical vitamin C deficiency and 15 % suffer from vitamin A de-ficiency. While every third child has a vitamin B6 deficiency, and every 10 has vitamin D deficiency in September and October, in April-May the number of children with insufficient provision of these vita-mins is 1.8-6.3 times more. А dynamic study of the vitamin provision for organized children found that half of them suffer from year-round vitamin B12 deficiency. Risk assessment of somatic health disorders in children with polyhipovitaminosis revealed that the low vitamin level in the blood leads to increase in the risk of children physical developmental abnormalities, dysfunction of vascular tone and vegetal re-sponsiveness, not to mention the fact that children's morbidity of chronic somatic diseases is 1.3–2.2 times more. Low vitamin provision increases 1.6–3.2 times the risk of physical and biological develop-ment during childhood; leads to the development of functional disorders of the respiratory, cardiovascu-lar and autonomic nervous system 2.1–3.3 times more. For children suffering from polyhipovitaminosis the risk of developing chronic diseases of the gastrointestinal tract and nervous system, allergic respira-tory and skin diseases, as well as musculoskeletal disorders is 1.3–4.1 times higher.

L∞ Observer for Uncertain Linear Systems

AbstractIn this paper, an L∞ observer design method is proposed for linear system subject to parameter uncertainty and bounded disturbance. The proposed L∞ observer, which satisfies a peak‐to‐peak disturbances attenuation performance, is designed to overbound the estimation error. Moreover, sufficient conditions for the design of L∞ observer are derived and expressed in terms of linear matrix inequalities (LMIs). The novelty of the proposed method is that we develop an L∞ observer that not only can attenuate bounded disturbance but also provides an upper bound of estimation error norm. Simulation results are presented to illustrate the effectiveness of the proposed method.