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Deep learning with specific network topologies has been successfully applied in many fields. However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates. [ABSTRACT FROM AUTHOR]

An adaptive fuzzy sliding control (AFSMC) approach is adopted in this paper to address the problem of angular position control and vibration suppression of rotary flexible joint systems. AFSMC consists of fuzzy-based singleton control action and switching control law. By adjusting fuzzy parameters with the self-learning ability to discard system nonlinearities and uncertainties, singleton control based on fuzzy approximation theory can estimate the perfect control law of feedback linearization. To enhance robustness, an additional switching control law is incorporated to reduce the approximation error between the derived singleton control action and the perfect control law of feedback linearization. AFSMC's closed-loop stability will be demonstrated via sliding surface and Lyapunov function analysis of error function. In order to demonstrate the effectiveness of the AFSMC in tracking performance as well as its ability to respond to model uncertainties and external perturbations, simulations are carried out using Simulink and Matlab in order to demonstrate how well it adapts to these situations. Based on these results, it can be concluded that the AFSMC performs satisfactorily in terms of tracking. [ABSTRACT FROM AUTHOR]

In this study, a robust fixed-time H∞ trajectory tracking controller for marine surface vessels (MSVs) is proposed based on self-structuring neural network (SSNN). First, a fixed-time H∞ Lyapunov stability theorem is proposed to guarantee that the MSV closed-loop system is fixed-time stable (FTS) and the L 2 gain is less than or equal to γ. This shows high accuracy and strong robustness to the approximation errors. Second, the SSNN is designed to compensate for the model uncertainties of the MSV system, marine environment disturbances, and lumped disturbances term constituted by the actuator faults (AFs). The SSNN can adjust the network structure in real time through elimination rules and split rules. This reduces the computational burden while ensuring the control performance. It is proven by Lyapunov stability that all signals in the MSV system are stable and bounded within a predetermined time. Finally, theoretical analysis and numerical simulation verify the feasibility and effectiveness of the control scheme. [ABSTRACT FROM AUTHOR]

This paper deals with Chebyshev wavelets. We analyze their properties computing their Fourier transform. Moreover, we discuss the differential properties of Chebyshev wavelets due to the connection coefficients. Uniform convergence of Chebyshev wavelets and their approximation error allow us to provide rigorous proofs. In particular, we expand the mother wavelet in Taylor series with an application both in fractional calculus and fractal geometry. Finally, we give two examples concerning the main properties proved. [ABSTRACT FROM AUTHOR]

This article studies the robust tracking control problems of Euler–Lagrange (EL) systems with uncertainties. To enhance the robustness of the control systems, an asymmetric tan-type barrier Lyapunov function (ATBLF) is used to dynamic constraint position tracking errors. To deal with the problems of the system uncertainties, the self-structuring neural network (SSNN) is developed to estimate the unknown dynamics model and avoid the calculation burden. The robust compensator is designed to estimate and compensate neural network (NN) approximation errors and unknown disturbances. In addition, a relative threshold event-triggered strategy is introduced, which greatly saves communication resources. Under the proposed robust control scheme, tracking behavior can be implemented with disturbance and unknown dynamics of the EL systems. All signals in the closed-loop system are proved to be bounded by stability analysis, and the tracking error can converge to the neighborhood near the origin. The numerical simulation results show the effectiveness and the validity of the proposed robust control scheme. [ABSTRACT FROM AUTHOR]

We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates' transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

We find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X , firstly in terms of finite linear combinations of functions of type K x i and then in terms of the projection π x n on span K x i i = 1 n , for random sequences of points x = x i i in X. Given a probability measure P , letting P K be the measure defined by d P K x = K x , x d P x , x ∈ X , our approach is based on the nonexpansive operator L 2 X ; P K ∋ λ ↦ L P , K λ ≔ ∫ X λ x K x d P x ∈ H , where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by H P , that is the operator range of L P , K . Our main result establishes bounds, in terms of the operator L P , K , on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type K x i , for x i i sampled independently from P , falls below a given threshold. For sequences of points x i i = 1 ∞ constituting a so-called uniqueness set, the orthogonal projections π x n to span K x i i = 1 n converge in the strong operator topology to the identity operator. We prove that, under the assumption that H P is dense in H , any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or L p norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H 2 D are presented as well. [ABSTRACT FROM AUTHOR]

The accurate prediction of the neutronic and thermal-hydraulic coupling system transient behavior is important in nuclear reactor safety analysis, where a large-scale nonlinear coupling system with strong stiffness should be solved efficiently. In order to reduce the stiffness and huge computational cost in the coupling system, the high-performance numerical techniques for solving delayed neutron precursor equation are a key issue. In this work, a new precursor integral method with an exponential approximation is proposed and compared with widely used Taylor approximation-based precursor integral methods. The truncation errors of exponential approximation and Taylor approximation are analyzed and compared. Moreover, a time control technique is put forward which is based on flux exponential approximation. The procedure is tested in a 2D neutron kinetic benchmark and a simplified high-temperature gas-cooled reactor-pebble bed module (HTR-PM) multiphysics problem utilizing the efficient Jacobian-free Newton–Krylov method. Results show that selecting appropriate flux approximation in the precursor integral method can improve the efficiency and precision compared with the traditional method. The computation time is reduced to one-ninth in the HTR-PM model under the same accuracy when applying the exponential integral method with the time adaptive technique. [ABSTRACT FROM AUTHOR]

For a class of nonlinear systems with a nonlinear relationship between input and output, a fuzzy control method combining interval type-2 and T-S fuzzy controller is proposed based on type-2 fuzzy system theory. In order to ensure its stability, anti-interference ability, and minimum approximation error, this design combines direct, indirect, supervised, and compensation control types to construct the controller. In this way, the structure of the controller not only has the characteristics of the type-2 fuzzy set, which can reduce the uncertainty of rules, but also has a T-S fuzzy model with linear combination of input variables, which can improve the modeling accuracy and reduce the number of rules of the system. By using the Lyapunov synthesis method, the global stability and the convergence of the closed-loop system under the condition that all variables are uniformly bounded are analyzed, and the adaptive laws of the system parameters are given as well. Finally, the effectiveness and superiority of the proposed method are verified by simulation. [ABSTRACT FROM AUTHOR]

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error. [ABSTRACT FROM AUTHOR]

Under U-model control design framework, a fixed-time neural networks adaptive backstepping control is proposed. The majority of the previously described adaptive neural controllers were based on uniformly ultimately bounded (UUB) or practical finite stable (PFS) theory. For neural networks control, it makes the control law as well as stability analysis highly lengthy and complicated because of the unknown ideal weight and unknown approximation error. Moreover, there has been very limited research focus on adaptive law for neural networks adaptive control in finite time. Based on fixed-time stability theory, a fixed-time bounded theory is proposed for fixed-time neural networks adaptive backstepping control. The most outstanding novelty is that fixed-time adaptive law for training weights of neural networks is proposed for fixed-time neural networks adaptive control. Furthermore, by combining fixed-time adaptive law and Lyapunov-based arguments, a valid fixed-time controller design algorithm is presented with universal approximation property of neural networks to ensure the system is fixed-time bounded, rather than PFS or UUB. The controller guarantees closed-loop system fixed-time bounded in the Lyapunov sense. The benchmark simulation demonstrated effectiveness and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this article, we establish some new Hermite–Hadamard-type inequalities involving the conformable fractional integrals. As applications, several inequalities for the approximation error in the midpoint formula and certain bivariate means are derived. [ABSTRACT FROM AUTHOR]

In this paper, we investigate singular Hadamard fractional boundary value problems. The existence and uniqueness of the exact iterative solution are established only by using an iterative algorithm. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have also been derived. [ABSTRACT FROM AUTHOR]

In order to improve the reconstruction precision and performance of 2D profiles, this paper presents a new high-precision extraction method of segment points of 2D profile features. Firstly, we extract the intervals which include initial segment points based on the curvature information of data points. Secondly, a grid is dynamically constructed in the interval and all the nodes of the grid are considered as candidate segment points for an optimization process. Thirdly, the optimization model of each feature curve is constructed on the basis of boundary constraints. Selection of the optimal segment point is related to two factors: the number of control points of B-spline and the total approximation error of all collected points to the fitted curves. Numerical experiments were conducted and the results demonstrate the efficacy of our method in capturing design intent and in industrial applications, compared to existing methods. [ABSTRACT FROM AUTHOR]

The complexity in the design of bipedal robots has motivated the use of simple mechanisms to accomplish the desired locomotion task with a minimum control effort. Nevertheless, a diverse set of conflictive design criteria must be met to develop the bipedal gait. In this paper, the synergy in the eight-bar mechanism design criteria to satisfy the bipedal lower-limb behavior is promoted by proposing a Pareto-based Nonlinear Mixed Discrete-Continuous Constrained Multiobjective Optimization Problem and by improving the search in the optimizer through the inclusion of the Multiselection Strategy into the multiobjective differential evolution algorithm, where the conflictive design objectives are the continuous path generation based on an approximation error to track twenty precision points of the bipedal gait and also the force transmission exerted when the foot reaches the ground. The manufacture of a prototype with a specific design trade-off experimentally validates the obtained synergistic design with the proposed design approach. In addition, the empirical analysis in simulation through a statistical validation indicates that the Multiselection Strategy explores and exploits the design solutions promoting the diversity, convergence, and capacity features of the obtained Pareto front with respect to other four multiobjective optimizers and consequently improves the reconfigurability in the design such that more alternatives result to the designer decision making. [ABSTRACT FROM AUTHOR]

In this study, it is proposed to define bivariate Chlodowsky variant of (p,q)-Bernstein-Stancu-Schurer operators. Therefore, Korovkin-type approximation theorems and the error of approximation by using full modulus of continuity are presented. Beside this, we introduce a generalization of the bivariate Chlodowsky variant of (p,q)-Bernstein-Stancu-Schurer operators and investigate its approximation in more general weighted space. Moreover, the numerical results are discussed in order to validate the accuracy of the bivariate Chlodowsky variant of (p,q)-Bernstein-Schurer operators. [ABSTRACT FROM AUTHOR]

The analysis of adaptors’ deformation and stress is an important prerequisite for the design of the adapters. This paper proposed the mathematical description and analytical method for the sponge cylinder and rubber cylinder under pressure. The analytical formulas on deformation and stress of interference fit between adapters and launchers are deduced. Also, the correctness of the analytical formula is verified by numerical simulation. Furthermore, the effect of precompression and the sponge’s thickness ratio on adapter’s deformation and stress is studied. The following results can be concluded: the results of numerical simulation are basically consistent with that of analytical formula. After installing the adapter into the launcher, the rubber layer will move inward as a whole, then the sponge layer and the rubber layer are subjected to radial and tangential compressive stress. When the precompression is smaller than 0.0075 or larger than 0.0015, the deformation and stress of adapter are approximately in proportion to precompression. If the precompression keeps constant, the deformation and stress of adapter will have extremum when the thickness ratio of sponge equals to a certain value. [ABSTRACT FROM AUTHOR]

A design of sliding mode controllers (SMC) with adaptive capacity is presented. This control technique is formed by two cascaded SMC controllers, one of them having an adaptive neural compensator (ANC); both are put on a WMR (wheeled mobile robot). The mobile robot is divided into a kinematics and a dynamics structure; the first SMC controller acts only on the kinematic structure and the SMC with neural adaptive compensator on the other one. The dynamic SMC was designed applying an inverse dynamic controller and using the model dynamics of the WMR. The adaptive neural compensation (ANC) was used in order to reduce the control error caused by the dynamics variations but it conveys a residual approximation error, so a sliding part was designed to cancel such error. This technique allows achieving the control objective despite parameter variations and external disturbances that take place in the dynamics; on the other hand, the ANC can adjust its neural parameters to reduce the dynamics variations of the WMR and thus improve the trajectory tracking control. Problems of convergence and stability are treated and design rules based on Lyapunov’s theorem are given. [ABSTRACT FROM AUTHOR]

In this paper, advanced interval type-2 fuzzy sliding mode control (AIT2FSMC) for robot manipulator is proposed. The proposed AIT2FSMC is a combination of interval type-2 fuzzy system and sliding mode control. For resembling a feedback linearization (FL) control law, interval type-2 fuzzy system is designed. For compensating the approximation error between the FL control law and interval type-2 fuzzy system, sliding mode controller is designed, respectively. The tuning algorithms are derived in the sense of Lyapunov stability theorem. Two-link rigid robot manipulator with nonlinearity is used to test and the simulation results are presented to show the effectiveness of the proposed method that can control unknown system well. [ABSTRACT FROM AUTHOR]

A fuzzy adaptive analytic model predictive control method is proposed in this paper for a class of uncertain nonlinear systems. Specifically, invoking the standard results from the Moore-Penrose inverse of matrix, the unmatched problem which exists commonly in input and output dimensions of systems is firstly solved. Then, recurring to analytic model predictive control law, combined with fuzzy adaptive approach, the fuzzy adaptive predictive controller synthesis for the underlying systems is developed. To further reduce the impact of fuzzy approximation error on the system and improve the robustness of the system, the robust compensation term is introduced. It is shown that by applying the fuzzy adaptive analytic model predictive controller the rudder roll stabilization system is ultimately uniformly bounded stabilized in the H-infinity sense. Finally, simulation results demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We show the interest of nonparametric methods taking into account the boundary correction techniques for a numerical evaluation of an approximation error between the stationary distributions of G/M/1 and M/M/1 queueing systems, when the density function of the general arrivals law G in the G/M/1 system is unknown and defined on a bounded support. To compute this error, we use two kinds of norms: the L1 norm and the weight norm. Numerical examples based on simulation studies are presented for the two cases of considered norms. A comparative study of the results has been provided. [ABSTRACT FROM AUTHOR]

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. The a priori and a posteriori rules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. [ABSTRACT FROM AUTHOR]

We consider a family of classification algorithms generated from a regularization kernel scheme associated with l¹-regularizer and convex loss function. Our main purpose is to provide an explicit convergence rate for the excess misclassification error of the produced classifiers. The error decomposition includes approximation error, hypothesis error, and sample error. We apply some novel techniques to estimate the hypothesis error and sample error. Learning rates are eventually derived under some assumptions on the kernel, the input space, the marginal distribution, and the approximation error. [ABSTRACT FROM AUTHOR]

We propose a decentralized error-bounded sliding mode control mechanism that ensures the prescribed tracking performance of a robot manipulator. A tracking error-transformed sliding surface was constructed and the barrier Lyapunov function (BLF) was used to ensure the transient and steady-state time performance of the positioning function of a robot manipulator as well as satisfy the ordinary sliding mode control properties. Unknown nonlinear functions and approximation errors are estimated by the RBF network and adaptive compensator. The effectiveness of the proposed control scheme was determined by comparing the results of an experiment evaluation with those of the conventional slidingmode control (SMC) and finite-time terminal slidingmode control (FTSMC) methods. [ABSTRACT FROM AUTHOR]

We study the density problem and approximation error of reproducing kernel Hilbert spaces for the purpose of learning theory. For a Mercer kernel K on a compact metric space (X, d), a characterization for the generated reproducing kernel Hilbert space (RKHS) HK to be dense in C(X) is given. As a corollary, we show that the density is always true for convolution type kernels. Some estimates for the rate of convergence of interpolation schemes are presented for general Mercer kernels. These are then used to establish for convolution type kernels quantitative analysis for the approximation error in learning theory. Finally, we show by the example of Gaussian kernels with varying variances that the approximation error can be improved when we adaptively change the value of the parameter for the used kernel. This confirms the method of choosing varying parameters which is used often in many applications of learning theory. [ABSTRACT FROM AUTHOR]

Based on the niche characteristics, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Considering the close degree of Niche as the consequent of adaptive T-S fuzzy control system, the hybrid control law is designed by tracking, continuous supervisory, and adaptive compensation. Adaptive compensator is used in the controller to compensate the approximation error of fuzzy logic system and the effect of the external disturbance. The adaptive law of consequent parameters, which is achieved in this paper, embodies system adaptability as biological individual. It is proved that all signals in the closedloop system are bounded and tracking error converges to zero by Lyapunov stability theory. The effectiveness of the approach is demonstrated by the simulation results. [ABSTRACT FROM AUTHOR]

This paper focuses on the fuzzy variable structure control for uncertain systems with disturbance. Specifically, the fuzzy control is introduced to estimate the control disturbance, the switching control is included to compensate for the approximation error, and they possess the characteristic of simpleness in design and effectiveness in attenuating the control chattering. Some typical numerical examples are presented to demonstrate the effectiveness and advantage of the fuzzy variable structure controller proposed. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present a general viscosity iteration process {xn} which defined by xn + 1 = (I - anA)Txn + ßn?f(xn) + (an - ßn)xn and to study the convergence of {xn}, where T is a nonexpansive mapping and A is a strongly positive linear operator, if {an}, {ßn} satisfy appropriate conditions, then iteration sequence {xn} converges strongly to the unique solution x* ∈ f(T) of variational inequality ((A-?f)x*,x-x*) = 0,forallx ∈ f(T). Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others. [ABSTRACT FROM AUTHOR]

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out. [ABSTRACT FROM AUTHOR]