Lipschitz Continuous Research Articles

The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection

The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection

Diffusive limits of Lipschitz functionals of Poisson measures

Diffusive limits of Lipschitz functionals of Poisson measures

Periodic self-triggered strategy for highly non-linear hybrid stochastic delayed networks based on impulsive control

This article investigates the stabilization issue of highly non-linear hybrid stochastic delayed networks (HSDNs) via periodic self-triggered control under impulse (PS-TCI). Firstly, the existence of a unique global solution for highly non-linear HSDNs under PS-TCI is studied. Then, a stabilization criterion for highly non-linear HSDNs is established, by combining a graph-theoretic approach with a novel Lyapunov-based analysis, based on a ‘genuine’ Lyapunov function defined by introducing an auxiliary timer. Therein, the less conservative polynomial growth condition and local Lipschitz condition for the drift and diffusion coefficients are used than the linear growth condition and global Lipschitz condition. Meanwhile, the design idea of PS-TCI is based on the evolution of an upper bound of the mathematical expectation for Lyapunov function (not directly Lyapunov function or system state), which implies that the triggered instant of PS-TCI is not a random variable. Finally the theoretical results are employed to study the stability of a class of FitzHugh–Nagumo circuits networks and the central pattern generators networks of a hexapod robot, and correlative numerical simulations are provided for demonstration.

Smooth compositions enhance safety of the fuzzy systems

Fuzzy models provide an empirical structure for controller design upon the experience of the operator, where the system output is delivered from a block box. Hence, there is a lack of effective way for assuring the safety specifications during the system transitions. This is to say, there are safety-oriented restrictions where the system stated are prohibited to enter under the fuzzy based controller paradigm, however, the system transitions are not visible to the designer. Therefore, the safety margin of the fuzzy controlled systems is limited, since, when data arriving to the closed loop system have considerable level of noises and uncertainties, then, the standard fuzzy models will have difficulties to face the unpredicted conditions of closed loop system. Hence, the current paper proposes that the employment of the smooth fuzzy models will assist highly in avoiding the states enter to the critical areas. In this regards, we have defined reachability diameter of the fuzzy model and proved that the fuzzy models with the so-called smooth fuzzy compositions can potentially have more safe functioning in the closed loop systems rather than standard fuzzy models due to the of the tighter reachability diameter and their Lipschitz continuity characters. It will result to their higher robust performance and this claim has been supported by different simulations of the examples.

Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator

Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

On Lipschitz Continuity and Smoothness Up to the Boundary of Solutions of Hyperbolic Poisson’s Equation

On Lipschitz Continuity and Smoothness Up to the Boundary of Solutions of Hyperbolic Poisson’s Equation

Existence and optimal controls of non-autonomous for impulsive evolution equation without Lipschitz assumption

In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.

A new kind of double phase elliptic inclusions with logarithmic perturbation terms II: Applications

This paper studies several special cases of a double phase elliptic inclusion problem (DPEI) that involves a nonlinear and nonhomogeneous partial differential operator with unbalanced growth and logarithmic perturbation terms, and two multivalued functions defined in the domain and its boundary. We establish existence and extremality results, focusing on the following two assumptions:• the multivalued terms are formulated by the Clarke’s generalized subdifferential operators of locally Lipschitz functions;• the multivalued terms are generated by discontinuous multifunctions.When the appropriate multivalued functions are formulated by two interval functions, we develop a unifying method to construct the nontrivial sub- and supersolutions for the inequalities and inclusions under considerations, hence we obtain suitable existence and extremality results.

Hölder continuous maps on the interval with positive metric mean dimension

Fix a compact metric space X with finite topological dimension. Let C0(X) be the space of continuous maps on X and Hα(X) the space of α-Hölder continuous maps on X, for α ∈ (0, 1]. Let H1(X) be the space of Lipschitz continuous maps on X. We have H1(X) ⊂ Hβ (X) ⊂ Hα (X) ⊂ C0 (X), where 0 < α < β < 1. It is well-known that if Φ ∈ H1 (H), then Φ has metric mean dimension equal to zero. On the other hand, if X is a manifold, then C0 (X) contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any α ∈ (0, 1), there exists Φ ∈ Hα ([0, 1]) with positive metric mean dimension.

Weak approximation schemes for SDEs with super-linearly growing coefficients

We propose a new class of weak approximation schemes for stochastic differential equations with coefficients of suplinearly growth. Both the modified weak Euler schemes and the drift-implicit weak Euler scheme are studied. Under certain non-globally Lipschitz conditions, the proposed schemes are proved to have first-order convergence in the weak sense. Numerical experiments are included to confirm the theoretical results.

Optimal control of three-dimensional unsteady partial differential equations with convection term in continuous casting

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application

The class of quasimonotone mappings are known to be more general and applicable than the classes of pseudomonotone and monotone mappings. However, only very few results can be found in the literature on quasimonotone variational inequality problems and most of these results are on weak convergent algorithms. In this paper, we study the quasimonotone variational inequality problem (VIP) with constraint of fixed point problem (FPP) of quasi-pseudocontractive mappings. We introduce a new inertial Tseng’s extragradient method with self-adaptive step size for approximating the minimum-norm solutions of the aforementioned problem in the framework of Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a common (minimum-norm) solution of the quasimonotone VIP and FPP of quasi-pseudocontractive mappings without the knowledge of the Lipschitz constant of the cost operator. We provide several numerical experiments for the proposed method in comparison with existing methods in the literature. Finally, we applied our result to image restoration problem. Our result improves, extends and generalizes several of the recently announced results in this direction.

Lipschitz-agnostic, efficient and accurate rendering of implicit surfaces

AbstractIn this paper, we propose an accurate and controllable rendering process for implicit surfaces with no or unknown analytic Lipschitz constants. Our process is built upon a ray-casting approach where we construct an adaptive Chebyshev proxy along each ray to perform an accurate intersection test via a robust and multi-stage searching method. By taking into account approximation errors and numerical conditions, our methods comprise several pre-conditioning and post-processing stages to improve the numerical accuracy, which potentially applied recursively. The intersection search is performed by evaluating a QR decomposition on the Chebyshev proxy function, which can be done in a numerically accurate way. Our process achieves comparable accuracy to other techniques that impose more constraints on the surface, e.g., knowledge of Lipschitz constants, and higher accuracy compared to approaches that impose similar constraints as our approach.

Controllability of nonlinear fractional integrodifferential systems involving multiple delays in control

This work studies the existence of solutions and approximate controllability of fractional integrodifferential systems with Riemann-Liouville derivatives and with multiple delays in control. We establish suitable assumptions to prove the existence of solutions. Controllability of the system is shown by assuming a range condition on control operators and Lipschitz condition on non-linear functions. We use the concepts of strongly continuous semigroup rather than resolvent operators. Finally, an example is give to illustrate the theory.

Numerical approximations of stochastic delay differential equations with delayed impulses

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example.

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Given any d-dimensional Lipschitz Riemannian manifold (M,g) with heat kernel p, we establish uniform upper bounds on p which can always be decoupled in space and time. More precisely, we prove the existence of a constant C>0 and a bounded Lipschitz function R:M→(0,∞) such that for every x∈M and every t>0, supy∈Mp(t,x,y)≤Cmin{t,R2(x)}−d/2.This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by (M,g). In the case ∂M≠0̸, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on ∂M.We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.

Dynamics analysis, synchronization and FPGA implementation of multiscroll Hopfield neural networks with non-polynomial memristor

The number of attractors in a memristor-based multiscroll Hopfield Neural Network (HNN) is typically coupled with the number of polynomials, which leads to a coupling between the computational complexity and resource utilization in circuit implementation. To decouple this relationship, we propose a non-polynomial memristor that satisfies the Lipschitz condition. Regardless of whether it is used to simulate synaptic behavior, simulate the impact of electromagnetic radiation, or a combination of both scenarios, it can conveniently control the generation of single-direction or multiple-direction multiscroll attractors without adding or reducing any terms. By constructing Lyapunov functions, the sufficient condition for these multiscroll memristor HNNs to be bounded is obtained. After improving the feasibility of linear matrix inequalities, a strongly adaptive observer is proposed. After uniting an adaptive sliding mode control method, we propose a new adaptive synchronization scheme to simulate neural network synchronization. Finally, the digital circuit implementation and functional verification of these memristor-based multiscroll HNNs are completed using a field-programmable gate array (FPGA). Based on this, an image encryption circuit is designed so that the FPGA can directly encrypt images and transmit them to the IO device.

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.