Lipschitz Continuous Research Articles

BOUNDEDNESS AND EXISTENCE ANALYSIS SOLUTION OF AN OPTIMAL CONTROL PROBLEMS ON MATHEMATICAL COVID-19 MODEL

The investigation given right here is part of a literature review on mathematical models that apply analytical mathematics. The present work focuses on the COVID-19 model, which incorporates optimum control variables previously investigated and interpreted by Hakim. Depending on the current model, we will further develop the analysis and demonstrate the non-negativity condition as well as the boundedness criteria for the solutions. Additionally, we conduct several supplementary analyses by applying the Lipschitz function to examine the uniqueness of the solutions and the existence of the solution are hold on the autonomous system. This work to supports the previously findings that incorporating an optimal control into the model can reduce COVID-19 treat on public. Finally, that research verifies that the control variables used in the research satisfy all of the existence criteria, as outlined in Theorem 5 of this work.

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

Iterative methods for solving constraint nonlinear monotone equations have been developed and improved by many researchers. The aim of this research is to present a modified three-term conjugate descent (TTCD) derivative-free method for constrained nonlinear monotone equations. The proposed algorithm requires low storage memory; therefore, it has the capability to solve large-scale nonlinear equations. The algorithm generates a descent and bounded search direction dk at every iteration independent of the line search. The method is shown to be globally convergent under monotonicity and Lipschitz continuity conditions. Numerical results show that the suggested method can serve as an alternative to find the approximate solutions of nonlinear monotone equations. Furthermore, the method is promising for the reconstruction of sparse signal problems.

Event‐triggered adaptive fault‐tolerant control for heterogeneous nonlinear multi‐agent systems

SummaryIn this paper, a distributed adaptive output feedback consensus tracking control scheme is developed for heterogeneous nonlinear multi‐agent systems (MASs) with actuator failures. Unlike most existing results, the parameters in the considered MAS are unknown, and the system's nonlinear functions are not required to satisfy the Lipschitz condition. A kind of K‐filters with an additional design parameter is designed to suppress the effects of unknown actuator failures. In addition, an event‐triggering mechanism with a switching threshold method is designed to reduce the communication burden. With the proposed control scheme, the tracking error of each subsystem converges exponentially to an adjustable bound. Finally, a nonlinear MAS consisting of multiple single‐link robot manipulators is given to verify its effectiveness.

A distributed algorithm with network‐independent step‐size and event‐triggered mechanism for economic dispatch problem

AbstractThe economic dispatch problem (EDP) poses a significant challenge in energy management for modern power systems, particularly as these systems undergo expansion. This growth escalates the demand for communication resources and increases the risk of communication failures. To address this challenge, we propose a distributed algorithm with network‐independent step sizes and an event‐triggered mechanism, which reduces communication requirements and enhances adaptability. Unlike traditional methods, our algorithm uses network‐independent step sizes derived from each agent's local cost functions, thus eliminating the need for detailed network topology knowledge. The theoretical derivation identifies a range of step size values that depend solely on the objective function's strong convexity and the gradient's Lipschitz continuity. Furthermore, the proposed algorithm is shown to achieve a linear convergence rate, assuming the event triggering threshold criteria are met for linear convergence. Numerical experiments further validate the effectiveness and advantages of our proposed distributed algorithm by demonstrating its ability to maintain good convergence characteristics while reducing communication frequency.

Feedback Stabilization of Quasi-One-Sided Lipschitz Nonlinear Discrete-Time Systems with Reduced-Order Observer

The feedback stabilization problem for nonlinear discrete-time systems with a reduced-order observer is investigated, in which the nonlinear terms of the systems satisfy the quasi-one-sided Lipschitz condition. First, a discrete-time reduced-order observer for nonlinear systems is designed. Then, a feedback controller with a reduced-order observer is designed for realizing the stabilization of nonlinear discrete-time systems. We prove that the design of a feedback controller and reduced-order observer of systems can be carried out independently in the case of discrete-time with nonlinear terms, which largely reduces the computational complexity of the observer and controller. The introduction of the quasi-one-sided Lipschitz condition simultaneously enhances the robustness and stability of nonlinear control systems. Finally, the feasibility and effectiveness of the proposed design approach is verified by a numerical simulation.

On the well-posedness of the Cauchy problem for the two-component peakon system in Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m,n)∈Ck(R)∩Wk,1(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m,n)\\in C^{k}(\\mathbb {R}) \\cap W^{k,1}(\\mathbb {R})$$\\end{document} with k∈N∪{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in \\mathbb {N}\\cup \\{0\\}$$\\end{document}. This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂tm(t,x)=∂x[m(t,x)(u(t,x)-∂xu(t,x))(u(-t,-x)+∂x(u(-t,-x)))],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t m(t,x)= \\partial _x[m(t,x)(u(t,x)-\\partial _xu(t,x)) (u(-t,-x)+\\partial _x(u(-t,-x)))], \\end{aligned}$$\\end{document}where m(t,x)=1-∂x2u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m(t,x)=\\left( 1-\\partial _{x}^2\\right) u(t,x)$$\\end{document}. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}. Moreover, we derive criteria for blow-up of the local solution in this class.

Maximal directional derivatives in Laakso space

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a [Formula: see text]-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a [Formula: see text]-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces, Carnot groups and Banach spaces. Hence, the techniques used in these spaces do not generalize to metric measure spaces.

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

On the integration of $$L^0$$-Banach $$L^0$$-modules and its applications to vector calculus on $$\textsf{RCD}$$ spaces

AbstractA finite-dimensional $$\textsf{RCD}$$ RCD space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $$L^0$$ L 0 -Banach $$L^0$$ L 0 -modules of independent interest. Roughly speaking, we study how to ‘patch together’ vector fields defined on the leaves that are measurable with respect to the foliation parameter.

An efficient self-adaptive algorithm for finding common solutions to pseudomonotone variational inequalities and split fixed point problems in Hilbert spaces

In this paper, we introduce a new self-adaptive algorithm for finding a solution to the system consisting of pseudomonotone variational inequalities and split fixed point problems. Our algorithm does not depend on the norm of the transfer mapping and the Lipschitz constant of the operators. In addition, the control parameters are self-adaptive. We also give some numerical examples to illustrate the effectiveness of the proposed algorithm and compare it with Lohawech et al.’s algorithm in Lohawech et al. (2018) and Nnakwe et al.’s algorithm in Nnakwe et al. (2021).

On Lipschitz solutions of mean field games master equations

We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player.

Yet another proof of the density in energy of Lipschitz functions

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian–Sobolev space). Our result covers first-order Sobolev spaces of exponent p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (1,\\infty )$$\\end{document}, defined over a complete separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal 1-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES

Abstract Let $({\mathcal{M}},d,\mu)$ be the metric measure space with a Dirichlet form $\mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${\mathcal{M}}$ to its Poisson integral Ptu on ${\mathcal{M}}\times\mathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].

Investigating integrodifferential equations associated with fractal–fractional differential operators

This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

On Some Problems with Multivalued Mappings

We consider some problems with a set-valued mapping, which can be reduced to minimization of a homogeneous Lipschitz function on the unit sphere. Latter problem can be solved in some cases with a first order algorithm—the gradient projection method. As one of the examples, the case when set-valued mapping is the reachable set of a linear autonomous controlled system is considered. In several settings, the linear convergence is proven. The methods used in proofs follow those introduced by B.T. Polyak for the case where Lezanski– Polyak–Lojasiewicz condition holds. Unlike algorithms that use approximation of the reachable set, the proposed algorithms depend far less on dimension and other parameters of the problem. Efficient error estimation is possible. Numerical experiments confirm the effectiveness of the considered approach. This approach can also be applied to various set-theoretical problems with general set-valued mappings.

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint

Abstract The starting point of this paper is the study of the asymptotic behavior, as p → ∞ {p\to\infty} , of the following minimization problem: min ⁡ { 1 p ⁢ ∫ Ω | ∇ ⁡ v | p + 1 2 ⁢ ∫ Ω ( v - f ) 2 , v ∈ W 1 , p ⁢ ( Ω ) } . \min\biggl{\{}\frac{1}{p}\int_{\Omega}|\nabla v|^{p}+\frac{1}{2}\int_{\Omega}(% v-f)^{2},\quad v\in W^{1,p}(\Omega)\biggr{\}}. We show that the limit problem provides the best approximation, in the L 2 {L^{2}} -norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the ( N - 1 ) {(N-1)} -Hausdorff measure of the jump set of the function.