Lipschitz Continuous Research Articles

Hybrid Nash Equilibrium Seeking Under Partial-Decision Information: An Adaptive Dynamic Event-Triggered Approach

This paper is concerned with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hybrid</i> Nash equilibrium (NE) seeking problem over a network in a partial-decision information scenario. Each agent has access to both its own cost function and local decision information of its neighbors. First, an adaptive gradient-based algorithm is constructed in a fully distributed manner with the guaranteed convergence to the NE, where the network communication is required. Second, in order to save communication cost, a novel event-triggered scheme, namely, edge-based adaptive dynamic event-triggered (E-ADET) scheme, is proposed with on-line-tuned triggering parameter and threshold, and such a scheme is proven to be fully distributed and free of Zeno behavior. Then, a hybrid NE seeking algorithm, which is also fully distributed, is constructed under the E-ADET scheme. By means of the Lipschitz continuity and the strong monotonicity of the pseudo-gradient mapping, we show the convergence of the proposed algorithms to the NE. Compared with the existing distributed algorithms, our algorithms remove the requirement on global information, thereby exhibiting the merits of both flexibility and scalability. Finally, two examples are provided to validate the proposed NE seeking methods.

Dynamics of multi-valued retarded p-Laplace equations driven by nonlinear colored noise

This paper mainly considers the long-term behavior of p-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved.

On the Robustness of Belief-Rule-Based Expert Systems

Belief rule base (BRB) expert system has been widely used in complex system modeling. Robustness is crucial to the modeling performance and safety of BRB. For a better understanding and utility of BRB, there is thereby an urgent need to know what kind of influence each part of BRB may have when the disturbance occurs. Aiming at this, a more comprehensive analysis of BRB robustness is conducted in this article. First, the Lipschitz condition for BRB is defined. With the definitions, a new robustness analysis method of BRB is proposed, which is conducted from four aspects: 1) the input transformation; 2) the matching degree calculation; 3) the matching degree normalization; and 4) the rule aggregation. Moreover, five guidelines for BRB construction are proposed by analyzing its robustness, which can offer a practical guide for users to establish, adjust, and improve the BRB model for specific applications. The robustness analysis of the BRB expert system for the relay health-state evaluation is conducted to verify the effectiveness of the proposed method.

Adaptive dynamic output feedback FTC for nonlinear systems with unknown nonlinearities and faults

AbstractThe problem of adaptive dynamic output feedback fault tolerant control (FTC) for a class of nonlinear systems with unknown nonlinearities and faults is investigated, in which the nonlinearities do not need to satisfy the Lipschitz condition, so the system can represent a wider range of nonlinear system models. An adaptive dynamic output feedback FTC is designed, where the output and its derivatives are fully applied to compensate the actuator faults on systems. The designed control method can effectively offset the effect of faults on systems so as to stabilized the nonlinear system. Furthermore, based on the designed controller, the coupling term can be eliminated reasonably and less conservative stability conditions for the closed‐loop system have been obtained. The proposed control strategy pledges that the states of the closed‐loop system are uniformly ultimately bounded (UUB). Two numerical examples are presented to exhibit the effectiveness and merits of the proposed control scheme.

Some characterizations of Lipschitz spaces via commutators of the Hardy-Littlewood maximal operator on slice spaces

Let M be the Hardy-Littlewood maximal operator and b be a locally integrable function. Denote by M_b and [b,M] the maximal commutator and the nonlinear commutator of M with b. In this paper, we give necessary and sufficient conditions for the boundedness of M_b and [b,M] on slice spaces when the function b belongs to Lipschitz spaces, by which a new characterization of non-negative Lipschitz functions is obtained.

Infinite horizon impulse control of stochastic functional differential equations driven by Lévy processes

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional L p -Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Fixed Point Theorems for Semigroups of Lipschitzian Mappings

The main purpose of this paper is to extend the results of fixed point theorems for lipschitzian semigroups. The proofs we give follow the results of Ishihara, Suantai and Puengrattana theorems. Using one of the proofs, we also develop a fixed point theorem result for nonempty asymptotically total mapping semigroups on uniformly convex Banach spaces.

On (α,p)-Cyclic Contractions and Related Fixed Point Theorems

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as (α,p)-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of (α,p)-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations.

Lipschitz Continuity Results for a Class of Parametric Variational Inequalities and Applications to Network Games

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks.

A collisionless singular discrete Cucker-Smale model with deterministic perturbations

In this paper, we investigate two non-linearly perturbed extensions of the discrete Cucker-Smale model with singular coupling weights. The first perturbation is that all agents have non-identical free-will accelerations, and the second is that all agents have identical intrinsic dynamics with the Lipschitz property. For the first model, we apply the induction method and discrete energy method to show that agents avoid collisions for any time and flocking occurs under some initial conditions, if the diameter of agents’ free-will accelerations is summable. For the second model, we obtain collision-avoiding flocking occurrence under suitable initial data and the Lipschitz constant of the function for the intrinsic dynamics. We also provide several numerical examples to illustrate our main results.

Mean square stability of the split-step theta method for non-linear time-changed stochastic differential equations

This paper investigates the split-step theta (SST) method to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the SST method is proved, and the SST method attains the classical 1 of convergence. In addition, the mean square stability of the time-changed stochastic differential equations is investigated. Two examples are presented to show the consistency of the theoretical results.

On the continuity of control allocation for surface vessels with two azimuth thrusters

This paper presents two novel results concerning the control allocation problem (CAP) for marine crafts with 2 azimuth thrusters. The first result is a novel closed form solution for the CAP which ensures Lipschitz continuity of the azimuth angle. The second result is a convex optimization reformulation of the CAP for the case of limited rate of changes for the actuators orientation which includes a novel regularization term that avoids anomalous behaviors due to underactuation. The effectiveness of the two methods are showcased through numerical simulations.

Cross-modal challenging: Projection of brain response on stereoscopic image quality ranking

This work presents a novel cross-modality method for acquiring human visual perception on stereoscopic image quality ranking (SIQR), aiming to learn latent biological representations and thinking tokens from brain activity. The core idea is to directly project electroencephalogram (EEG) information onto a SIQR neural network model, rather than simulate the complex neuronal connections. To the end, we establish a multimodal brain-visual dataset, and propose a Cross-GAN model including a binocular attention-based fusion (BAF) image modality encoder, a multiscale spatiotemporal (MST) EEG modality encoder and a multimodal feature generation GAN (MF-GAN). In the BAF image encoder, the progressive interactive attention fusion is designed to highlight the significant regions of the image from monocular view to binocular view, which contributes to the global information description of single view and fused view. In the MST EEG encoder, spectral norm normalization constraint is adopted to assure the Lipschitz continuity, and the multiscale convolution blocks is presented to capture EEG representations from spatiotemporal perspective. Finally, the monomodal image and EEG features are sent to the MF-GAN for creating the optimal brain-visual manifold under the control of the generative loss function. Extensive experiments on the brain-visual multimodality SIQR database prove that the CrossGAN can improve the performance via projecting brain response onto image evaluation model in a cross-modal manner, with no need of EEG trails in application.

Split Lipschitz-type functions

In recent years, Lipschitz-type functions have been studied in an extensive manner. Particularly, we have locally Lipschitz functions, uniformly locally Lipschitz functions and Lipschitz in the small functions, etc. To answer some questions related to upper semicontinuity, recently in 2019, a new class of functions, called split continuous, is defined which is weaker than that of continuous functions. In this article, we define ‘split’ versions of Lipschitz-type functions. Various interesting properties of these functions are also investigated.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator

In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms.

Stability of grating diffraction problems for plane wave incidence: Explicit dependence on wavenumbers and incident angles

Suppose a time-harmonic plane wave is incident onto an impenetrable grating profile of Dirichlet or Impedance kind in two dimensions. The grating profile is assumed to be given by a Lipschitz function. We derive a stability estimate of this grating diffraction problem using the variational method with a transparent boundary condition. An explicit dependence of solutions on the incident wavenumber and the incident angle is obtained, which carries over to transmission problems for penetrable gratings. Our approach relies heavily on using Rellich's identities in periodic structures.

Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces

Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces

Wong-Zakai approximations for stochastic differential equations with path-dependent coefficients

In this paper, we shall prove an L p ( p &gt; 2 ) L^{p}(p&gt;2) strong convergence rate of the non-adapted Wong-Zakai approximation for stochastic differential equations with path-dependent coefficients under d ∞ d_{\infty } -Lipschitz conditions.