Lipschitz Continuity Condition Research Articles

Inertial relaxed CQ algorithm for split feasibility problems with non-Lipschitz gradient operators

In this paper, we propose a new inertial relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces. The main advantages of the proposed algorithm are that the introduced stepsize is bounded away from zero and relaxation parameter sequences imposed on the stepsize itself are removed. These features help accelerate our method. Also, a weak convergence theorem for our method is established without Lipschitz continuous and firmly-nonexpasive conditions of split feasibility problem related gradient and projection mappings, respectively. As an application, we consider multiple-sets split feasiblity problems. Finally, some preliminary numerical experiments are provided for illustration and comparison.

Iterative Learning Fuzzy Control for Nonlinear Systems with Adaptive Gain and without Resetting Condition

In this paper, we present two ILC schemes. The first one is a PD-type iterative learning control with an initial state algorithm to solve the trajectory tracking problem for nonlinear systems with uncertainties and without satisfying the classical resetting condition. -norm method is used to prove the asymptotic stability of the closed loop system and the simulation results on perturbed nonlinear system have been given. The second approach is a simple P-type iterative learning fuzzy control scheme to solve the trajectory tracking problem for MIMO nonlinear systems. The control design is applicable to deal with a class of nonlinear systems without satisfying the global Lipschitz continuity condition, for which a fuzzy logic term is added to cope with unknown parameters. In addition, the swarm optimization algorithm is used to design the optimum iterative learning fuzzy control (ILFC). Using Lyapunov theory, the asymptotic stability of the closed loop system is guaranteed over the whole finite time. Finally, an illustrative example on two-link manipulator is provided to illustrate the effectiveness of the proposed controller.

A nonlinear version of the distributed Halanay inequality and its application

In this paper, first we prove a nonlinear version of Halanay inequality with distributed delay. Then, we use it to establish a local exponential stability result for a problem appearing in neural networks theory under the name Hopfield neural network system. This is done for activation functions violating the usual standard condition of Lipschitz continuity. One of the activation functions may be non‐Lipschitz continuous. Finally, an example is provided to illustrate the developed theory.

Strong convergence of the forward–backward splitting algorithms via linesearches in Hilbert spaces

Many real-world problems in applied sciences, engineering and economics can be reformulated as the convex minimization problem of two objective functions. In order to solve this problem, the forward–backward splitting algorithm and its modifications have been invented and used to investigate the convergence analysis. In this work, we discuss strong convergence of the sequence generated by the forward–backward algorithm involving the viscosity approximation method. The stepsizes studied in this paper are defined by two different kinds of linesearches which do not require the Lipschitz continuity condition on the gradient. Finally, we provide numerical experiments to show the efficiency of the proposed methods in signal recovery. It is reported that our algorithms have a good convergence behavior in practice without information of Lipschitz constants of the gradient of functions.

English

In this paper, we propose a smoothing Levenberg-Marquardt method for the symmetric cone complementarity problem. Based on a smoothing function, we turn this problem into a system of nonlinear equations and then solve the equations by the method proposed. Under the condition of Lipschitz continuity of the Jacobian matrix and local error bound, the new method is proved to be globally convergent and locally superlinearly/quadratically convergent. Numerical experiments are also employed to show that the method is stable and efficient.

The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The Lp-norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

Event‐triggered adaptive fixed‐time output feedback fault tolerant control for perturbed planar nonlinear systems

AbstractThis paper considers an event‐triggered adaptive fixed‐time output feedback fault tolerant control problem for a class of uncertain planar nonlinear systems, in which the actuator has an unknown drift fault and the loss of effectiveness fault. Firstly, by using the adaptive backstepping technique and event‐triggered theory, a constructive event‐triggered adaptive fixed‐time state feedback controller is presented under a generalized Lipschitz continuous condition. Secondly, for the unmeasured state, a continuous fixed‐time observer is constructed. It is shown that the global practically fixed‐time stability of the closed‐loop systems is ensured by the bilimit homogeneity technique and Lyapunov theory analysis. Moreover, the system is demonstrated by contradiction to be nonzeno. Finally, the fan speed control system is constructed to demonstrate the validity of the proposed strategy and the application of the general systems.

Exponential convergence of a proximal projection neural network for mixed variational inequalities and applications

This paper proposes a novel proximal projection neural network (PPNN) to deal with mixed variational inequalities. It is shown that the PPNN has a unique continuous solution under the condition of Lipschitz continuity and that the trajectories of the PPNN converge to the unique equilibrium solution exponentially under some mild conditions. In addition, we study the influence of different parameters on the convergence rate. Furthermore, the proposed PPNN is applied in solving nonlinear complementarity problems, min–max problems, sparse recovery problems and classification and feature selection problems. Finally, numerical and experimental examples are presented to validate the effectiveness of the proposed neurodynamic network.

An analysis of solutions to fractional neutral differential equations with delay

This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag–Leffler functions; their stability properties are hence studied by using use Rouché’s theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified.

Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative

Abstract The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q, where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

Robust and resilient stabilization and tracking control for chaotic dynamical systems with uncertainties

In this article, a robust resilient design methodology for stabilization and tracking control for a class of chaotic dynamical systems is proposed. In particular, a resilient quasi-sliding mode control law is formulated to suppress the chaotic behaviour of these dynamical systems, while considering uncertainties in control input. By using Lyapunov stability theory, sufficient conditions are derived in terms of Linear Matrix Inequalities (LMIs) for a more general class of chaotic nonlinear dynamical systems satisfying the Lipschitz continuity condition. The control gains are obtained via LMI technique in the presence of bounded additive uncertainties in control input. Numerical simulations are provided for Chua’s circuit, Lorenz system, and 4D Lu hyper-chaotic system to show the supremacy and effectiveness of the proposed design methodologies.

A proximal neurodynamic model for solving inverse mixed variational inequalities

This paper proposes a proximal neurodynamic model (PNDM) for solving inverse mixed variational inequalities (IMVIs) based on the proximal operator. It is shown that the PNDM has a unique continuous solution under the condition of Lipschitz continuity (L-continuity). It is also shown that the equilibrium point of the proposed PNDM is asymptotically stable or exponentially stable under some mild conditions. Finally, three numerical examples are presented to illustrate effectiveness of the proposed PNDM.

Asymptotic behavior of a BAM neural network with delays of distributed type

In this paper, we examine a Bidirectional Associative Memory neural network model with distributed delays. Using a result due to Cid [J. Math. Anal. Appl.281(2003) 264–275], we were able to prove an exponential stability result in the case when the standard Lipschitz continuity condition is violated. Indeed, we deal with activation functions which may not be Lipschitz continuous. Therefore, the standard Halanay inequality is not applicable. We will use a nonlinear version of this inequality. At the end, the obtained differential inequality which should imply the exponential stability appears ‘state dependent’. That is the usual constant depends in this case on the state itself. This adds some difficulties which we overcome by a suitable argument.

A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.

New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.

Global Fixed-Time Output Feedback Stabilization of Perturbed Planar Nonlinear Systems

The global fixed-time stabilization of a class of perturbed planar nonlinear systems using output feedback is addressed in this brief. Under a generalized Lipschitz continuous condition, a constructive controller design of global output feedback is proposed based on a continuous fixed-time observer. The bi-limit homogeneity technique is employed to show global fixed-time stability of the closed-loop system. Simulation of a MEMS mirror model illustrates the appealing performance of the proposed method.

A study of the local convergence of a fifth order iterative method

We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Fréchet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.

An inertial projection neural network for solving inverse variational inequalities

A novel inertial projection neural network (IPNN) is proposed for solving inverse variational inequalities (IVIs) in this paper. It is shown that the IPNN has a unique solution under the condition of Lipschitz continuity and that the solution trajectories of the IPNN converge to the equilibrium solution asymptotically if the corresponding operator is co-coercive. Finally, several examples are presented to illustrtae the effectiveness of the proposed IPNN.

Iterative learning control for nonlinear differential inclusion systems

SummaryIn this paper, we propose an iterative learning control strategy to track a desired trajectory for a class of uncertain systems governed by nonlinear differential inclusions. By imposing Lipschitz continuous condition on a set‐valued mapping described by a closure of the convex hull of a set and using D‐type and PD‐type updating laws with initial iterative learning, we establish the iterative learning process and give a new convergence analysis with the help of Steiner‐type selector. Finally, numerical examples are provided to verify the effectiveness of the proposed method with suitable selection of set‐valued mappings. An application to the speed control of robotic fish is also given.

Local convergence of parameter based method with six and eighth order of convergence

This paper dealt with the local convergence study of the parameter based sixth and eighth order iterative method. This analysis discuss under assumption that the first order Frechet derivative satisfied the Lipschitz continuity condition. In this way, we also proposed the theoretical radius of convergence of these methods. Finally, some numerical examples demonstrate that our results apply to compute the radius of convergence ball of iterative method to solve nonlinear equations. We compare the results with the method in Kumar et al. (J Comput Appl Math 330:676–694, 2018) and observe that by our approach we get much larger balls as existing ones.