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In this article, being different from conventional time-discretization (simply called discretization) formulas, explicit linear left-and-right 5-step (ELLR5S) formulas with sixth-order precision are proposed. The general sixth-order ELLR5S formula with four variable parameters is developed first, and constraints of these four parameters are displayed to guarantee the zero stability, consistence, and convergence of the formula. Then, by choosing specific parameter values within constraints, eight specific sixth-order ELLR5S formulas are developed. The general sixth-order ELLR5S formula is further utilized to generate discrete zeroing neural network (DZNN) models for solving time-varying linear and nonlinear systems. For comparison, three conventional discretization formulas are also utilized. Theoretical analyses are presented to show the performance of ELLR5S formulas and DZNN models. Furthermore, abundant experiments, including three practical applications, that is, angle-of-arrival (AoA) localization and two redundant manipulators (PUMA560 manipulator and Kinova manipulator) control, are conducted. The synthesized results substantiate the efficacy and superiority of sixth-order ELLR5S formulas as well as the corresponding DZNN models.

The linearity assumption is widely used when acquiring the hydrodynamic coefficients of a floating structure. However, the linear damping is frequently underestimated, especially for the natural frequency. To investigate the sloping seafloor effects on the damping terms of a single module of a semi-submersible Very Large Floating Structure (VLFS), this paper revisits the conventional formulation and further proposes the direct integration method for obtaining the linear and quadratic damping coefficients from free-decay tests. Numerical free-decay simulations of the single module over variable bathymetry are carried out by the CFD numerical tank. Corresponding model tests are also implemented to verify and validate against the numerical solutions. The effects of the sloping seafloor, as well as the water depth, on the hydrodynamic coefficients are investigated based on the validated CFD modeling. Both numerical and experimental results indicate that the acquisition of the linear and quadratic damping coefficients is sensitive to the data-processing and identification approaches. For the case studied in present paper, the identification errors introduced by the conventional method are 1.5 % while they are 0.5 % using the direct integration method. The quadratic damping coefficient for heave mode decreases about 10.4 % when the sloping angle increases from 0 to 6 deg.

Fractional order nonlinear evolution equations have emerged in recent times as being very important model for depicting the interior behavior of nonlinear phenomena that exist in the real world. In particular, Schrodinger-type fractional nonlinear evolution equations constitute an aspect of the field of quantum mechanics. In this study, the (2 + 1)-dimensional time-fractional nonlinear Schrodinger equation and (1 + 1)-dimensional time-space fractional nonlinear Schrodinger equation are revealed as having different and novel wave structures. This is shown by constructing appropriate analytic wave solutions. A successful implementation of the advised rational ( 1 / ϕ ′ ( ξ ) ) -expansion method generates new outcomes of the considered equations, by comparing them with those already noted in the literature. On the basis of the conformable fractional derivative, a composite wave variable conversion has been used to adapt the suggested equations into the differential equations with a single independent variable before applying the scheme. Finally, the well-furnished outcomes are plotted in different 3D and 2D profiles for the purpose of illustrating various physical characteristics of wave structures. The employed technique is competent, productive and concise enough, making it feasible for future studies.

In this article, we study the finite-time stabilization and the asymptotic stabilization with probability one of Markovian jump Boolean control networks (MJBCNs) by sampled-data state feedback controls (SDSFCs). Based on the semi-tensor product (STP), we introduce an augmented variable multiplied by the vector form of the switching signal and the state of MJBCN. We find that under SDSFC, the sequence of the states of the augmented variable at sampling instants satisfies the Markov property. Based on the convergences of the switching signal and the augmented variable, we obtain the sufficient and necessary criteria for the finite-time stabilization and the asymptotic stabilization of MJBCNs by SDSFCs, respectively. Moreover, for the two kinds of stabilization, the feedback matrices of SDSFCs are constructed, respectively. Finally, the obtained results are applied to an apoptosis network and a model of the lactose operon in the Escherichia Coli.

In this study, we investigate the (2 + 1)-dimensional Korteweg-De Vries (KdV) equation with the extension of time-dependent coefficients. A symbolic computational method, the simplified Hirota’s method, and a long-wave method are utilized to create various exact solutions to the suggested equation. The symbolic computational method is applied to create the Lump solutions and periodic lump waves. Hirota’s method and a long-wave method are implemented to explore single-, double- and triple-M-lump waves, and interaction physical phenomena such as an interaction of single-M-lump with one-, two-soliton solutions, as well as a collision of double-M-lump with single-soliton waves. Furthermore, the simplified Hirota’s method is employed to explore complex multi-soliton solutions. To realize dynamics, the gained solutions are drawn via utilizing different arbitrary variable coefficients.

Classical bearing capacity theory was developed mainly based on spatially uniform soil properties, which cannot account for the influence of inherent soil variability. If the soil strength is heterogeneous, then using the average strength may overestimate the bearing capacity of foundations, because the failure mechanism may preferentially mobilise the weaker soils. In this study the aim is to establish a theoretical model using upper-bound solutions applied to the bearing capacity analysis of shallow foundations on undrained clay considering spatial variability. The model is derived on the principle of least energy dissipation using a four-parameter variation on Prandtl's mechanism. The theoretical model developed is verified by the random finite-element (FE) method in spatially varying soil conditions. The results show that the model can accurately capture the effect of spatially varying strength on the shallow foundation failure mechanism. The difference of bearing capacity factor between the proposed model and the FE model is within 5%, which demonstrates that the four-parameter model has an accuracy that is comparable to FE analysis with many hundreds of degrees of freedom. Another advantage of the theoretical model is that the possible non-convergence in FE analysis can be avoided, and hence, the calculation efficiency is significantly enhanced. The model is therefore suitable for rapid quantification of bearing capacity in spatially varying soils.

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by "simple" arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable., 26 pages. Notations have been improved. Final version to appear in Journal of Number Theory

The article presents a new approach for optimizing the trajectories of spacecraft with finite thrust. The recently proposed angular variable – the auxiliary longitude – is considered as an independent variable. The advantages of using this variable to optimize a trajectory with a fixed angular distance and free transfer duration are shown. A complete system of necessary optimality conditions for the considered representation of the optimal control problem is presented. An approach to solving the minimum-fuel problem is considered that is based on the sequential solution of the problems of optimizing the limited power trajectories and the trajectories with a constant exhaust velocity. This approach includes verifying the existence of a solution to the minimum-fuel problem by solving the thrust minimization problem. We present a method for optimization of the finite-thrust trajectories with constant exhaust velocity, based on the application of the maximum principle and the continuation method, which does not require the user to specify an initial guess for the initial values of costate variables. The article presents numerical examples to illustrate the application of such approach, and discusses the properties of the obtained optimal trajectories.

In this study, the generalized modified variable-coefficient KdV equation with external-force term (gvcmKdV) describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained. Moreover, several new solutions have been constructed for the gvcmKdV. Additionally, the classical direct similarity reduction method is used to reduce the gvcmKdV to a nonlinear ordinary differential equation. Building on the solutions given in the previous literature for the reduced equation, many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.

In this paper we study the Controllability and the generalized Ulam-Hyers stability of Riemann-Liouville fractional order linear differential equations with variable coefficients in Banach spaces. We obtained the desired results by means of fixed point principle and fractional calculus. Furthermore, we used the Gronwall’s inequality to investigate above type of stability for Riemann-Liouville fractional order linear differential equation with time dependent coefficients under some specific assumptions. An example is also provided at the end to illustrate the theoretical work.

In this paper, the high-order nonlinear system is commonly studied in an underactuated weakly coupled mechanical system, the control design is difficult from the tractional control design for nonlinear systems. Thus, we study the finite-time fuzzy adaptive error constraint control problem for stochastic high-order nonlinear nonstrict feedback systems. Fuzzy logic systems (FLSs) are utilized to identify the unknown nonlinear dynamics, a new error transfer variable is used to achieve the prescribed performance. Based on adding a power integrator technique and adaptive backstepping recursive control, a novel adaptive fuzzy finite-time prescribed performance control scheme is developed. By utilizing stochastically finite-time stable theory, the proposed control method can guarantee that the high-order system is semi-global finite-time stable in probability (SGFSP). Finally, both numerical and practical simulations are provided to verify the feasibility and effectiveness of the developed control method.

This paper provides a unified formulation of optimal guidance-to-collision law for a target with an arbitrary acceleration or deceleration. The collision course for general target acceleration or deceleration is first determined from the engagement geometry in conjunction with the nonlinear engagement kinematics in the proposed approach. The heading error de_ned in the collision course is then adopted as a variable to be nullified for accomplishing the intercept condition. The proposed guidance law is derived based on the heading error dynamics and the optimal error dynamics to ensure optimality and finite-time convergence. As illustrative examples, the proposed guidance command for a constant target acceleration and a target deceleration in the form of a quadratic function of speed are provided. Additionally, the time-to-go prediction method is suggested for implementing the proposed method. The characteristics of the proposed guidance command are analytically investigated to provide insight into the proposed method. The key benefits of the proposed method lie in not producing unnecessary guidance commands near a target compared to other methods and ensuring optimality in guidance command even in the nonlinear engagement kinematics. Finally, numerical simulations are performed to validate the proposed method and to show our findings.

This article proposes a novel gain-scheduling control method to improve vehicle lateral stability based on a variable-parameter-dependent approach. A time-varying velocity-dependent model to describe the vehicle lateral dynamic characteristics is constructed under cornering stiffness uncertainties and controller saturation. A novel Lyapunov function with variable-parameter-dependent Lyapunov matrix is established to develop the conditions. Using the proposed Lyapunov function, a general condition of the variable-parameter-dependent saturated robust yaw moment controller is provided in terms of time-varying parameter-dependent matrix inequalities. In order to obtain a tractable solution, a condition is further developed with finite linear matrix inequalities. Moreover, an optimal distribution method is adopted to generate the desired yaw moment based on torque allocation. These torques are computed by optimizing the distribution errors and tire workloads. Simulations results under both J-turn and double-lane changing scenarios are used to illustrate the merits of the proposed method. The control synthesis approach is also applicable to other applications involving time-varying parameters.

Multi-term variable-order fractional differential equations (VO-FDEs) are considered to be one of the tools to illustrate the behavior of transient-regime real-life phenomena precisely. The analytical solutions of VO-FDEs are generally very hard to obtain. Therefore, in this study, an approximation method for the solution of multi-term VO-FDEs is proposed using shifted Chebyshev polynomials of the third kind (SCP3). The method employs the operational matrices of SCP3 to proximate the VO-FDEs with a system of algebraic equations. The solution of which also yields the approximate result for the multi-term VO-FDE. An error analysis is also included. The SCP3 method is examined on several examples demonstrating the precision and prowess of the method.

With the historical review, it can determine that plasma scientists devolved many numerical schemes in literature to solve the kinetic plasma models such as Vlasov- Maxwell system, but the fatal problem is that most of the designed algorithms violate the conservation laws. This issue is significant and cannot disregard because the Vlasov Maxwell system is connected with conservations properties and can evaluate after performing some effective mathematical operations. In this prospective, we formulated the Atangana-Baleanu fractional derivative with Caputo concepts based fractional-order plasma model to study the fractional performance of the plasma particles and further suggested a higher-order conservative numerical algorithm that follows the conservation laws, based on operational matrices theory which is evaluated by using the concepts of Shifted Gegenbauer estimations. The numerical convergence of the designed algorithm is also inspected to validate the competency and compatibility. The comprehensive analysis of the fractional performance of the plasma particles is also part of the study. Moreover, this concept can extend to the variable order and multi-dimensional problems for Vlasov and Boltzmann system.

In this paper, we are interested to explore stable configurations of $d$-dimensional gravastars constructed from the interior $d$-dimensional de Sitter and exterior $d$-dimensional Schwarzschild/Reissner-Nordstr\"om (de Sitter) black holes through cut and paste approach. We consider the linearized radial perturbation preserving the original symmetries to explore their stability by using three different types of matter distributions. The resulting frameworks represent unstable structures for barotropic, Chaplygin, and phantomlike models for every considered choice of exterior geometries. However, matter contents with variable equations of state have a remarkable role to maintain the stability of gravastars. We conclude that stable structures of gravastars are obtained only for generalized variable models with exterior $d$-dimensional Schwarzschild/Reissner-Nordstr\"om-de Sitter black holes., Comment: 21 pages, 12 figures, to appear in Chinese J. Physics

This article is mainly devoted to the investigation on the stability and Hopf bifurcation of fractional-order neural networks with mixed delays. Applying a suitable substitution of variable, a novel equivalent fractional-order neural networks concerning single delay is set up. By analyzing the corresponding characteristic equation of the involved fractional-order delayed neural networks and choosing the time delay as bifurcation parameter, we derive a new sufficient condition to guarantee the stability behavior and the appearance of Hopf bifurcation for the considered fractional-order delayed neural networks. The study reveals that the time delay is a key factor which has a vital impact on stability and Hopf bifurcation of neural networks. The obtained results of this work can be effectively applied to design neural networks. The numerical simulations and bifurcation diagrams are displayed to verify the rationality of the analytical results.

The aim of this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on variable exponent Morrey spaces of an integral form. As an application of the boundedness of the maximal operator, we establish Sobolev-type inequalities for Riesz potentials $$I_{\alpha (\cdot )}f$$ of variable order $$\alpha (\cdot )$$ of functions f in variable exponent Morrey spaces of an integral form.

Fractional differential equations with constant coefficients can be readily handled by a number of methods, but those with variable coefficients are much more challenging. Recently, a method has appeared in the literature for solving fractional differential equations with variable coefficients, the solution being in the form of an infinite series of iterated fractional integrals. In the current work, we consider fractional differential equations with Atangana–Baleanu integro-differential operators and continuous variable coefficients, and establish analytical solutions for such equations. The representation of the solution is given by a uniformly convergent infinite series involving Atangana–Baleanu operators. To the best of our knowledge, this is the first time that explicit analytical solutions have been given for such general Atangana–Baleanu differential equations with variable coefficients. The corresponding results for fractional differential equations with constant coefficients are also given.

In this article, a general form of multiterm variable-order fractional delay differential equations (GVOFDDEs) is presented. The introduced GVOFDDEs have variable-order multi-terms and integer-order derivatives as well for all terms delayed or with normal argument. The variable order derivative is a generalization of fractional and integer orders, so it considered in this work. The collocation approach is applied with the aid of shifted Chebyshev polynomials to solve the presented GVOFDDEs as a matrix discretization technique. The presented technique transforms all terms of GVOFDDEs to a matrix equation with novel operational matrices. The qualification of the presented scheme is measured by many numerical test examples.

Variable Reluctance (VR) resolver is an angular position sensor that is commonly used in the automotive industry. In this paper, A general method for determining the winding configuration of VR resolvers is proposed, that eases the manufacturing process. At first, the operational principle of conventional winding has been presented using mathematical formulations. Following that, the non-overlapping tooth coil winding method, as the ideal winding option for VR resolver, has been extended that can be applied to the VR resolver with arbitrary poles and stator tooth numbers. The method of determining the reference and signal windings presented and used to determine the winding for a 2, 6, and 8 poles resolvers. The validity of the proposed winding is investigated using 3D Time Stepping Finite Element Analysis (3D-TSFEA) and experimental measurements.

Fixed-time stability (FXTS) and fixed-time control (FXTC) of dynamic systems are reconsidered in this article based on special functions from the view of improving the estimate accuracy of settling time (ST) and reducing the chattering caused by the sign function. First, by means of the idea of contradiction and variable transformations, some generic FXTS criteria are established and some upper bounds of ST are directly calculated and expressed by several special functions. It is further proved that these estimates are the most accurate compared with the existing results. Besides, to suppress the chattering caused by the sign function, some saturation functions are constructed to replace the sign function and the FXTS of the new system obtained by replacing is ensured by rigorous theoretical analysis. As applications, the problem of stabilization for chaotic systems in fixed or preassigned time is explored. Especially, an innovative saturation controller is developed to realize preassigned-time stabilization, where the convergence time is prescribed in advance according to actual requirement and the control gains are finite, the existing control methods with time-varying infinite gains are essentially improved. Lastly, three numerical examples are provided to verify the improved estimates of ST and the chattering reduction.

This paper focus on model predictive control (MPC) design in the context of sampled-data control systems with full-state measurements. It is shown that recent results on this area can be successfully generalized to cope with sampled-data MPC. The open-loop plant is subjected to polytopic parameter uncertainty and at sampling times a controlled output variable satisfies a set of convex constraints. A guaranteed H2 performance index with infinity horizon is minimized such as the feedback control preserves asymptotic stability and feasibility. The design conditions are expressed though differential linear matrix inequalities (DLMIs). Continuous-time systems are treated with no kind of discrete-time modeling approximation. Comparisons with classical methods from the literature dealing with continuous-time systems are presented and discussed. Examples are included for illustration.

This article addresses the H∞ control issue for Takagi-Sugeno (T-S) fuzzy systems via an event-triggered control mechanism. First, a hybrid event-triggered control strategy combined with an adaptively adjusted approach is proposed, by which the data transmission can be effectively reduced, and meanwhile, the Zeno behavior is excluded. Then, a discontinuous Lyapunov function is constructed for the resulting hybrid event-triggered control system, and a novel lemma is developed for studying the exponential stability and disturbance attenuation performance. By utilizing a time-varying variable to manage the error during the interevent intervals together with using the newly proposed lemma, sufficient criteria are established for the stability and control design of the event-triggered T-S fuzzy system with the imperfect premise matching. Finally, numerical simulations are presented to illustrate the usefulness and advantages of the obtained theorems.

Many industrial applications result in the constrained underdetermined nonlinear system (UNS) that needs to be solved. The existing solutions are reported to the constrained UNS with time-invariant coefficients, and they may not perform well for the time-variant case. In this paper, we attempt to address the above limitation by providing a new discrete-time zeroing neural network (DTZNN) for solving the time-variant UNS (TVUNS) under bound constraint. Specifically, the bound-constrained TVUNS (BC-TVUNS) is first transformed into a mixed nonlinear system by introducing a nonnegative variable. Then, a continuous-time ZNN (CTZNN) model is established to solve such a mixed system as well as the BC-TVUNS. By employing a special difference formula to discretize the CTZNN model, the new DTZNN model is thus proposed to obtain the solution of the BC-TVUNS. Theoretical analysis and comparative numerical results are presented to validate the effectiveness of the proposed DTZNN model over the previous DTZNN models. The DTZNN application potential is further indicated on the basis of the simulations on a dual-arm redundant robot using the proposed model.

The main target of our research is to achieve the solution to a variable-order fractional mobile-immobile advection-dispersion equation within the Atangana-Baleanu (AB) fractional operator. This equation is the best tool for modeling dynamical systems. The presented scheme is based on the collocation method and Lagrange polynomials. The given approach uses derivative operational matrices of integer and variable orders. The operational matrix of variable-order derivative in the AB sense is computed based on Lagrange polynomials in the presented study. The key theory of this approach is to convert the corresponding problem to a system of algebraic equations. The solution to the problem under study can be computed by utilizing the mentioned system and the collocation points. The aim of this paper is to show that this technique is easily used to solve fractional partial differential equations with excellent accuracy in results. To demonstrate the accuracy and efficiency of the suggested method, some numerical examples are provided.

As an effective paradigm to address uncertainty in payoffs and transition probabilities, robust stochastic games have been formulated in the literature. This paper is concerned with the computation of subgame perfect equilibria in stationary strategies (SSPEs) in robust stochastic games. To tackle this problem, we develop in this paper a globally convergent differentiable path-following method by exploiting the structures of the games. Incorporating a logarithmic-barrier term into each player’s payoff function with an extra variable between zero and one, we constitute a logarithmic-barrier robust stochastic game in which each player solves in each state a convex optimization problem. An application of the optimality conditions to the barrier game together with a fixed-point argument yields a polynomial equilibrium system for the barrier game. As a result of this system, we establish the existence of a smooth path that starts from an arbitrary mixed strategy profile and ends at an SSPE as the extra variable descends from one to zero. As an alternative scheme, we make up a convex-quadratic-penalty robust stochastic game and attain a globally convergent convex-quadratic-penalty differentiable path-following method for SSPEs in robust stochastic games. Numerical comparisons show that the logarithmic-barrier path-following method significantly outperforms the convex-quadratic-penalty path-following method. To further evince the value of the proposed methods, we apply the logarithmic-barrier path-following method to solve a supply chain configuration problem and a market entry problem from medical waste recycling.

In this brief, we contribute to adaptive neural network (NN) backstepping control design for the nonlinear double-integrator fractional-order (FO) systems comprising unknown dynamics and disturbances by the event-triggered procedure. To this end, a new triggering rule with a dynamical variable is first introduced, including some available static triggering rules as its particular form. Then, by using the contradiction and some properties of the Mittag-Leffler function, it is proved for the first time that the applied dynamical variable is positive and bounded for the FO systems. Hence, the inter-event time between any two successive triggering moments can be enlarged than static event-triggered results. By integrating the backstepping approach into the NN, an adaptive NN controller is constructed, which introduces a new analysis viewpoint on the Zeno phenomenon avoidance for the FO controllers. Under this controller, the tracking error converges to a small compact set including the origin. Finally, an example is given to show the feasibility of the proposed method.

As a special case of the Volterra system, the second-order Volterra (SOV) filter is very efficient for nonlinear system identification. The improved correntorpy based on the generalized Gaussian density function has been proven robust against impulsive noise. In this brief, we propose several SOV filters based on a recursive maximum correntropy (RMC) algorithm for nonlinear system identification. We first introduce a basic RMC algorithm, which faces a trade-off between filtering accuracy and tracking capability due to the use of a fixed forgetting factor (FFF). Two RMCs with variable FF (VFF) are further proposed to enhance the tracking ability. Simulation results demonstrate that our proposed algorithms outperform existing ones in impulsive noise environments and/or in time-varying systems.

This paper aims to propose an evidential reasoning (ER) rule considering the parameter uncertainty. As the essential parameters, the evidence weight and reliability make the ER rule constitute a generalized reasoning framework. Theoretically, the weight is affected by subjective cognition, while the reliability mainly reflects objective variation. However, most of the recent researches have focused on the quantitative calculation methods, which makes the differences in the property of the two parameters ignored. In this work, a relatively different idea from previous studies is provided, in which multi-source uncertainty of parameters is fully considered. On the one hand, the weight is profiled by interval variable with lower and upper bounds. On the other hand, the reliability is modeled by random variable with probability distribution function (PDF). Then, a unified inference model for evidence aggregation is developed based on the inference process of the ER rule. In addition, some basic properties of the model are clearly presented to illustrate the rationality of parameter uncertainty. Finally, a practical example is given to show the potential applications of the proposed model.

The nonlinear filtering problem is a hot spot in robot navigation research. As an advanced method that can effectively improve the robustness and accuracy of the system, the progressive Gaussian approximate filter with variable step size (PGAFVS) still has some shortcomings, how to resolve the nonlinear filtering problem in the application of tightly coupled integration under the premise of the prior uncertainty and further promote robustness high measurement accuracy, which becomes the purpose of this paper. This paper formulates the processing of trajectory tracking measurement noise problem as a Kalman filtering procedure and the measurement noise covariance matrix in controller, is jointly estimated based on the progressive Gaussian approximate filter (PGAF), after that, PGAFVS can be deduced. Then we proposed an adaptive fuzzy and backpropagation neural network controller based on PGAFVS (AFNPGA-VS) that can improve the application of tightly coupled integration under the premise of the prior uncertainty and further promote robustness high measurement accuracy. The simulation results show that the proposed algorithm outperforms the state-of-the-art methods.

In this paper, a chaotic strategy-based quadratic opposition-based learning adaptive variable-speed whale optimization algorithm is proposed. The improved algorithm is used to solve the problems that the whale optimization algorithm’s convergence accuracy and convergence speed are insufficient. Firstly, the proposed algorithm is initialized by a method based on chaotic maps and quadratic opposition-based learning strategy to obtain a population with better ergodicity. Secondly, by introducing an adaptive variable speed adjustment factor, each search link unites to form a negative feedback regulation network, thereby effectively balancing the algorithm’s exploration ability and exploitation ability. Finally, 20 benchmark test functions and 3 complex constrained engineering optimization problems were used to conduct extensive tests on the improved algorithm. The results show that the improved algorithm has better performance than others in terms of convergence speed and convergence accuracy in a majority of cases, and can effectively jump out of the local optimum.

The finite-time formation control of second-order linear multi-agent systems subject to constraints on both relative positions and relative velocities is studied in this paper. A novel barrier function-based sliding mode control (BFSMC) method, in which the relative state constraints are embedded through the barrier function, is proposed. The boundedness of sliding variable guarantees the satisfaction of the relative state constraints, and the convergence of the sliding variable to zero ensures that the formation will be reached in finite time. The singularity in controller is avoided. The effectiveness of the results is illustrated through a numerical example.

The fitted mesh numerical method is presented to solve parabolic problems involving an interior layer. A uniform mesh was applied in the temporal direction and fitted mesh type on the space variable yields the discrete problem. The convergence of the method is established and shown to the second-order in the temporal direction with the first-order convergent in the spatial one. Further, the scheme is proven to be parameter independent convergent and also confirmed by numerical experiments. In a nutshell, novelty of the present scheme is uniformly convergent and gives a more accurate solution than some existing methods in the literature.

The constant kernel width of generalized maximum correntropy criteria (GMCC) has arisen that the steady-state error and convergence speed can be mutually exclusive. To solve this problem, this paper proposes the variable kernel width (VKW) GMCC algorithm. Actually, due to the censored problem, the output data value beyond the limit of the recording device can not be well observed. In this case, we further developed a variable kernel width GMCC algorithm based on censored regression (CRVKWGMCC). Simulation results show that the proposed CR-VKWGMCC algorithm has excellent performance in both Gaussian and non Gaussian noise.

Stationarity is fundamental for time-series modeling and prediction. In this article, we focus on the radial basis function network-based autoregressive (RBF-AR) models which have been widely used in practical applications. Compared to previous work, we give a less-restrictive sufficient condition for the asymptotic stationarity of the RBF-AR model. The parameter estimation of the RBF-AR model is converted to the optimization of a variable projection functional with constraints of stationarity to always derive a stationary model. The constrained evolutionary algorithm is used to solve the optimization problem. Numerical results demonstrate the effectiveness of the proposed method.

We introduce a coordinate transformation of independent variable, such that the second-order nonlinear singularly perturbed boundary value problem (SPBVP) in the transformed coordinate is less stiff within the boundary layer. An initial value problem for a new dependent variable can be derived easily through the variable transformation. While the zero initial values are given, an unknown terminal value of the new variable at the right end is determined iteratively. We propose the modifications of the asymptotic solution and the uniform approximate solution of the SPBVP; hence, the modified analytic solutions can exactly satisfy both the boundary conditions at two ends. Some examples confirm that the novel methods can achieve better analytic and numerical solutions of the nonlinear SPBVP.

This work presents a high-order Bernstein-Bezier finite element (FE) discretisation to accurately solve time harmonic elastic wave problems on unstructured triangular mesh grids. Although high-order FEs possess many advantages over standard FEs, the computational cost of matrix assembly is a major issue in high-order computations. A key ingredient to address this drawback is to resort to low complexity procedures in building the local high order FE matrices. This is achieved in this work by exploiting the tensorial property of Bernstein polynomials and applying the sum factorisation method for curved elements. An efficient implementation of the analytical rules for affine elements is also proposed. Furthermore, element-level static condensation of the interior degrees of freedom is performed to reduce the memory requirements. Additionally, the applicability of the method with a variable polynomial order, based on a simple a priori indicator, is investigated. The computational complexities of sum factorisation, analytical rules and standard quadrature are first evaluated, in terms of the CPU time against the polynomial order. The analysis shows that the achieved numerical complexities compare favourably to those expected theoretically. A significant runtime saving is also obtained by using analytical rules and sum factorisation. The performance of the Bernstein-Bezier FEs is then assessed on various benchmark tests, over a wide range of frequencies. Results from the elastic wave scattering problem demonstrate the effectiveness of this method in coping with the pollution error, and its accuracy in resolving high order evanescent wave modes. Additionally, a wave transmission problem with high wave speeds contrast and a curved interior interface is considered, where a simple a priori indicator is proposed to assign the variable polynomial order. The study provides evidence of the great benefit of a non uniform p -refinement in reducing the computational cost and enhancing accuracy.

In this paper, we derive delay and fractional-order dependent sufficient conditions for fractional-order gene regulatory networks (FOGRNs) with time-varying delays to be stable with H ∞ /passivity performance. Distinct from the existing works on delayed FOGRNs, the Lyapunov-Krasovskii functional (LKF) is suitably structured so that the delay functions can be non-differentiable or even have jump type discontinuities. Further, the positive-definiteness condition is relaxed for the matrices involved in LKF. A new set of linear matrix inequality (LMI) conditions that ensure the stability of FOGRNs with certain H ∞ /passivity performance level is derived using various results on fractional derivatives/integrals and convex property of LMIs. Finally, the results obtained are verified with existing FOGRNs via numerical simulations.

This article presents an engineering method to optimize the stiffener outline shape and cross-sectional sizes simultaneously for stiffened plate structures. A new optimization model for stiffened plate is established, and a hierarchical algorithm including external and internal optimization module is proposed to tackle the shape and size variables respectively. In external optimization, genetic algorithm (GA) is utilized to optimize binary-coded shape variables, and when calculating individual fitness values in GA, an internal optimization only involving size variables is carried out. For each individual, the stiffener outline shape is determined by a given shape variable vector and a reasonable finite element model could be automatically built with a parametric modeling method. The solution of internal optimization problem can be obtained by size optimization method, and an offset vector which updates during each cycle is created to modify the relative position of plate and stiffeners, ensuring the practicality of the optimal results in engineering. The presented approach is firstly verified through a typical example and then it is applied to the structural design of a micro-satellite. The results show that the method is effective in dealing with the simultaneous outline shape and size optimization problem of stiffeners, and the optimal results could be directly applied in practical engineering structures.

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust.

This paper investigates solvability of fully coupled systems of forward–backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

Second-order optimality conditions for fuel-optimal control problems with both endpoints on manifolds are studied in this paper. Each candidate extremal from Pontryagin’s maximum principle is embed...

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

This paper is mainly concerned with the stability of inertial delayed neural networks (IDNNs) with stochastic delayed impulses. In contrast with previous related works, the stochastic impulsive intensity and density, and impulsive delays are considered simultaneously. Firstly, by appropriate variable substitution, the considered IDNNs model is converted to a first order differential equation with stochastic delayed impulsive effect. Secondly, by using the matrix measure approach and some stochastic analysis techniques, the stability criteria of IDNNs with stochastic delayed impulses are presented. Finally, numerical examples are provided to illustrate the obtained theoretical results.

This article proposes a new approach for computing a semiexplicit form of the solution to a class of traffic flow problems encoded by a Hamilton-Jacobi (HJ) partial differential equation (PDE), with time-switched Hamiltonian. Using a characterization of the problem derived from viability theory, we show that the solution associated with the problem can be formulated as a minimization problem involving the trajectory of an auxiliary dynamical system. A generalized Lax-Hopf formula for the switched Hamiltonian problem is derived, which enables us to compute the solution associated with affine initial or boundary conditions as a linear program involving the control function of the auxiliary dynamical system. This formulation allows us to compute the solution to the original problem exactly, unlike dynamic programming methods. In addition, this method allows one to very efficiently recompute the boundary conditions associated with an initial condition problem, allowing large-scale variable speed limit traffic control problems to be solved.

We introduce the concept of kinetic maximal L p -regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal L μ p -regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal L μ p -regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.