Lipschitz Continuous Research Articles

Caputo fractional backward stochastic differential equations driven by fractional Brownian motion with delayed generator

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions.

Time‐optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion with Clarke's subdifferential

In this article, we discuss the optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion of order with Clarke's subdifferential. Firstly, we establish the existence of mild solutions for this class of equations by employing the Krasnoselskii's fixed‐point theorem, without using Lipschitz conditions. Subsequently, we develop set of sufficient assumptions, utilizing Cesari property and Filippov theorem, to ensure the existence of feasible pairs for the feedback control systems. Eventually, using the Mazur theorem and Fatou's lemma, we derive the optimal state–control pair. The study also introduces and explores the problem of time‐optimal feedback control. In conclusion, an application is provided to enhance the theoretical findings.

Orbital Lipschitzian mappings and semigroup actions on metric spaces

In this paper we study some results on common fixed points of families of mappings on metric spaces by imposing orbit Lipschitzian conditions on them. These orbit Lipschitzian conditions are weaker than asking the mappings to be Lipschitzian in the traditional way. We provide new results under the two classic approaches in the theory of fixed points for uniformly Lipschitzian mappings: the one under the normal structure property of the space (which can be regarded as the Cassini-Maluta's approach) and the one after the Lifschitz characteristic of the metric space (Lifschitz's approach). Although we focus on the case of semigroup of mappings, our results are new even when a mapping is considered by itself.

Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T

Let $U$ be the set of admissible controls, $T>0$ and it be given a scale of Banach spaces $W[0;\tau]$, $\tau\in(0;T]$, such that the set of constrictions of functions from $W=W[0;T]$ to a closed segment $[0;\tau]$ coincides with $W[0;\tau]$; $F[\cdot;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. For the operator equation $x=F[x;u]$, $x\in W$, we introduce a comparison system in the form of functional-integral equation in the space $\mathbf{C}[0;T]$. We establish that, under some natural hypotheses on the operator $F$, the preservation of the global solvability of the comparison system pointed above is sufficient to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation. This fact itself is analogous to some results which were obtained by the author earlier. The central result of the paper consists in a set of signs for stable global solvability of functional-integral equations mentioned above which do not use hypotheses similar to local Lipschitz continuity of the right-hand side. As a pithy example of special interest, we consider a nonlinear nonstationary Navier–Stokes system in the space $\mathbb{R}^3$.

Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation

In this paper, we study three numerical schemes for the McKean-Vlasov equation {dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,∀t∈[0,T],μt is the probability distribution of Xt, where X0 : (Ω, F, ℙ) → (ℝd, B(ℝd)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends previous work [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] established in a one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme inspired by the K-means clustering). Two simulations are presented at the end of this paper: Burgers equation introduced in [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] and the network of FitzHugh- Nagumo neurons (see [J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, J. Math. Neurosci. 2 (2012) 1–50.] and [M. Bossy, O. Faugeras and D. Talay, J. Math. Neurosci. 5 (2015) 1–23.]) in dimension 3.

Convergence analysis of an explicit method and its random batch approximation for the McKean–Vlasov equations with non-globally Lipschitz conditions

In this paper, we present a numerical approach to solve the McKean–Vlasov equations, which are distribution-dependent stochastic differential equations, under some non-globally Lipschitz conditions for both the drift and diffusion coefficients. We establish a propagation of chaos result, based on which the McKean–Vlasov equation is approximated by an interacting particle system. A truncated Euler scheme is then proposed for the interacting particle system allowing for a Khasminskii- type condition on the coefficients. To reduce the computational cost, the random batch approximation proposed in [S. Jin, L. Li and J. Liu, J. Comput. Phys. 400 (2020) 108877.] is extended to the interacting particle system where the interaction could take place in the diffusion term. An almost half order of convergence is proved in Lp sense. Numerical tests are performed to verify the theoretical results.

Analyzing the Semilocal Convergence of a Fourth-Order Newton-Type Scheme with Novel Majorant and Average Lipschitz Conditions

Analyzing the Semilocal Convergence of a Fourth-Order Newton-Type Scheme with Novel Majorant and Average Lipschitz Conditions

Level sets of nonsmooth functions, Part 1: Lipschitz and piecewise-differentiable Rank Theorems

We present a piecewise-differentiable (PCr) Rank Theorem and extend a previously stated Lipschitz Rank Theorem, with the goal of characterizing the level sets of nonsmooth functions f:Rn→Rm. When the appropriate conditions are satisfied by the generalized derivatives of f, the Rank Theorems allow us to express a given level set f−1(c)⊂Rn locally as the graph of a nonsmooth function, within a homeomorphic transformation of the same class as f. We define PCr and Lipschitz submersions, immersions and maps of constant rank in terms of the most general conditions under which the corresponding Rank Theorems are applicable. Moreover, we develop sufficient conditions that are more easily verifiable for practical applications and relate them to existing full-rank conditions from the literature.

Some new estimates of integral inequalities and their applications

UDC 517.9, 517.928 We obtain several new integral inequalities in terms of fractional integral operators for the functions whose first derivatives satisfy either the conditions of the Lagrange theorem or the Lipschitz condition. In some special cases, the results obtained provide better upper estimates than those known in the literature for Bullen-type inequality and Hadamard-type right-hand side inequality. Finally, some error estimates for the trapezoidal formula are discussed.

The Truncated EM Method for Stochastic Differential Equations Driven by Fractional Brownian Motion

We mainly focus on the numerical method of fractional Brownian motion in this paper. On the basis of the numerical method of general SDEs, an approximation scheme is obtained for the stochastic differential equations about fractional noise. And we get it by using the Lipschitz condition and combining with the truncation function f∆ and g∆. Furthermore, we also prove the moment boundedness and convergence of the solution by some lemma. At last, we apply this method to the Gilpin-Ayala model. The orbital image of the solution and the form of numerical solution are given. The error of solution also has been simulated by MATLAB.

EMG-Based Automatic Gesture Recognition Using Lipschitz-Regularized Neural Networks

This article introduces a novel approach for building a robust Automatic Gesture Recognition system based on Surface Electromyographic (sEMG) signals, acquired at the forearm level. Our main contribution is to propose new constrained learning strategies that ensure robustness against adversarial perturbations by controlling the Lipschitz constant of the classifier. We focus on nonnegative neural networks for which accurate Lipschitz bounds can be derived, and we propose different spectral norm constraints offering robustness guarantees from a theoretical viewpoint. Experimental results on four publicly available datasets highlight that a good tradeoff in terms of accuracy and performance is achieved. We then demonstrate the robustness of our models, compared with standard trained classifiers in four scenarios, considering both white-box and black-box attacks.

Typical Lipschitz images of rectifiable metric spaces

Abstract This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into R m \mathbb{R}^{m} for m ≥ n m\geq n . For example, if E ⊂ R k E\subset\mathbb{R}^{k} , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n \mathcal{H}^{n} -almost everywhere and, if m > n m>n , preserves the Hausdorff measure of 𝐸. In general, we provide sufficient conditions, in terms of the tangent norms of 𝐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 𝐸, up to some constant multiple. Almost optimal results for strongly 𝑛-rectifiable metric spaces are obtained. On the other hand, for any norm | ⋅ | \lvert\,{\cdot}\,\rvert on R m \mathbb{R}^{m} , we show that, in the space of 1-Lipschitz functions from ( [ − 1 , 1 ] n , | ⋅ | ∞ ) ([-1,1]^{n},\lvert\,{\cdot}\,\rvert_{\infty}) to ( R m , | ⋅ | ) (\mathbb{R}^{m},\lvert\,{\cdot}\,\rvert) , the H n \mathcal{H}^{n} -measure of a typical image is not bounded below by any Δ > 0 \Delta>0 .

Observer‐based fault diagnosis for autonomous systems

AbstractWith the advancement in sensor and communication technology, autonomous systems have been incrementally reshaping the execution of tasks in commercial and military sectors. Since the systems are designed to complete tasks without or with minimal human intervention, fault diagnosis based on sensor data has been crucial to preventing accidents and fatalities. In this paper, fault diagnosis for autonomous systems is designed based on nonlinear adaptive observers, tested in numerical simulations, and implemented in a robotic platform. To this end, we utilize the persistence of excitation conditions on the parametric model of the faults. We derive sufficient conditions for the nonlinear adaptive observer in terms of linear matrix inequality to ensure the convergence of the estimates. Furthermore, we consider one‐sided Lipschitz conditions to obtain less conservative results. The main advantage of using the nonlinear adaptive observer is that the method converges quickly to the actual fault and requires minimum computational effort. However, solving the linear matrix inequality might not be trivial. Numerical simulations based on a single‐link flexible joint robot model and experimental tests in an autonomous quadcopter are performed to validate the effectiveness of the proposed method.

First-Order Algorithms Without Lipschitz Gradient: A Sequential Local Optimization Approach

Most first-order methods rely on the global Lipschitz continuity of the objective gradient, which fails to hold in many problems. This paper develops a sequential local optimization (SLO) framework for first-order algorithms to optimize problems without Lipschitz gradient. Operating on the assumption that the gradient is locally Lipschitz continuous over any compact set, SLO develops a careful scheme to control the distance between successive iterates. The proposed framework can easily adapt to the existing first-order methods, such as projected gradient descent (PGD), truncated gradient descent (TGD), and a parameter-free variant of Armijo linesearch. We show that SLO requires [Formula: see text] gradient evaluations to find an ϵ-stationary point, where Y is certain compact set with [Formula: see text] radius, and [Formula: see text] denotes the Lipschitz constant of the i-th order derivatives in Y. It is worth noting that our analysis provides the first nonasymptotic convergence rate for the (slight variant of) Armijo linesearch algorithm without globally Lipschitz continuous gradient or convexity. As a generic framework, we also show that SLO can incorporate more complicated subroutines, such as a variant of the accelerated gradient descent (AGD) method that can harness the problem’s second-order smoothness without Hessian computation, which achieves an improved [Formula: see text] complexity. Funding: J. Zhang is supported by the MOE AcRF [Grant A-0009530-04-00], from Singapore Ministry of Education. M. Hong is supported by NSF [Grants CIF-1910385 and EPCN-2311007]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2021.0029 .

About the general chain rule for functions of bounded variation

We give an alternative proof of the general chain rule for functions of bounded variation (Ambrosio and Maso, 1990), which allows to compute the distributional differential of φ∘F, where φ∈LIP(Rm) and F∈BV(Rn,Rm). In our argument we build on top of recently established links between “closability of certain differentiation operators” and “differentiability of Lipschitz functions in related directions” (Alberti et al., 2023): we couple this with the observation that “the map that takes φ and returns the distributional differential of φ∘F is closable” to conclude.Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.

Hardy decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Adversarial Robustness of Neural Networks From the Perspective of Lipschitz Calculus: A Survey

We survey the adversarial robustness of neural networks from the perspective of Lipschitz calculus in a unifying fashion by expressing models, attacks and safety guarantees, that is, a notion of measurable trustworthiness, in a mathematical language. After an intuitive motivation, we discuss algorithms to estimate a network’s Lipschitz constant, Lipschitz regularisation techniques, robustness guarantees, and the connection between a model’s Lipschitz constant and its generalisation capabilities. Afterwards, we present a new vantage point regarding minimal Lipschitz extensions, corroborate its value empirically and discuss possible research directions. Finally, we add a toolbox containing mathematical prerequisites for navigating the field (Appendix).

Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications

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To provide this result, we will need to prove that ρE(zx,zy)≤|z|ρE(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _E(zx,zy) \\le |z| \\rho _E(x,y)$$\\end{document} for x,y∈BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x,y \\in B_E$$\\end{document} under some conditions on z∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z \\in \\mathbb {C}$$\\end{document}. Lipschitz continuity of x↦(1-‖x‖2)Rf(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto (1-\\Vert x\\Vert ^2) \\mathcal {R}f(x)$$\\end{document} will yield some applications on interpolating sequences for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} which also extends classical results from B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document} to B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document}. Indeed, we show that it is necessary for a sequence in BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document} to be separated to be interpolating for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} and we also prove that any interpolating sequence for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} can be slightly perturbed and it remains interpolating.

Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions

On spaces of finite signed Borel measures on a metric space one has introduced the Fortet-Mourier and Dudley norms, by embedding the measures into the dual space of the Banach space of bounded Lipschitz functions, equipped with different – but equivalent – norms: the FM-norm and the BL-norm, respectively. The norm of such a measure is then obtained by maximising the value of the measure when applied by integration to extremal functions of the unit ball. We introduce Lipschitz extension operators, essentially based on those defined by McShane, and investigate their properties. A remarkable one is that non-trivial extreme points are mapped to non-trivial extreme points of FM- and BL-norm unit balls. Using these extension operators, we define suitable ‘small’ subsets of extremal functions that are weak-star dense in the full set of extreme points of the unit ball, for any underlying metric space. For connected metric spaces, we additionally find a larger set of extremal functions for the BL-norm, similar to such a set that was defined previously by J. Johnson for the FM-norm. This set is then also weak-star dense in the extremal functions. These results may open an avenue to obtaining computational approaches for the Dudley norm on signed Borel measures.

Stochastic Bregman extragradient algorithm with line search for stochastic mixed variational inequalities

In this paper, we present a stochastic Bregman extragradient algorithm with line search for solving a class of stochastic mixed variational inequality, which not only does not require any information of the Lipschitz constant but allows to search a potentially larger step size per iteration. Compared with the existing algorithms, the proposed algorithm allows different step sizes in the prediction and correction steps, thus enhancing the algorithm's flexibility. Under the generalized monotonicity, we derive the almost sure convergence, the iteration complexity O ( 1 / ϵ ) and the oracle complexity O ( 1 / ϵ 2 ) for our algorithm. Furthermore, under the generalized strong monotonicity and the sample size increases at a geometric rate, the linear convergence rate of the proposed algorithm with respect to the Bregman distance between the iterations and solution is established. Numerical results demonstrate a favourable comparison of the proposed algorithm with existing ones.