scholarly journals Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fang Lu ◽  
Chun-Rong Chen
Keyword(s):  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Normed Spaces ◽  
Lipschitz Property ◽  
Nonlinear Scalarization ◽  
Dual Spaces ◽  
Lipschitz Constants ◽  
Primal And Dual ◽  
Vector Equilibrium ◽  
The One

Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

scholarly journals Slice convergence of parametrised sums of convex functions in non-reflexive spaces

1999 ◽  
Vol 60 (3) ◽  
pp. 429-458 ◽  
Author(s):  
Robert Wenczel ◽  
Andrew Eberhard
Keyword(s):  
Convex Functions ◽  
Mosco Convergence ◽  
Normed Spaces ◽  
Unified Framework ◽  
Dual Spaces ◽  
Slice Convergence ◽  
Sequential Continuity ◽  
Kuratowski Convergence ◽  
Primal And Dual ◽  
Fenchel Conjugation

The objectives of this study of slice convergence are two-fold. The first is to derive results regarding the passage of certain semi–convergences through Young–Fenchel conjugation. These semi–convergences arise from the splitting of the usual slice topology in the primal and dual spaces into (non-Hausdorff) topologies: the upper slice topology ; a topology generating a convergence closely resembling the bounded–weak* upper Kuratowski convergence; along with the respective primal and dual lower Kuratowski topologies. This gives rise to topological convergences not reliant on sequentially–based definitions found in many such studies, and associated topological continuity results for conjugation (in normed spaces), in contrast to the usual sequential continuity exhibited by analogues of Mosco convergence. The second objective is to study the passage of slice convergence through addition. Such sum theorems have been derived in other works and we establish previous theorems from a unified framework as well as obtaining a new result.

Adaptive Observer of One-Sided Lipschitz Nonlinear Systems

2013 ◽  
Vol 336-338 ◽  
pp. 417-422
Author(s):  
Hao Liu ◽  
Jia Chen Ma ◽  
Ming Li Yang ◽  
Yu De Sun
Keyword(s):  
Nonlinear Systems ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Adaptive Observer ◽  
The Past ◽  
Lipschitz Nonlinear Systems ◽  
The One ◽  
Adaptation Law ◽  
Almost All

Aiming to design an adaptive observer for one-sided Lipschitz nonlinear systems, the paper focuses on extensions to the restrictive nature of the Lipschitz continuity condition and the conservativeness of approaches proposed in the past. Main contributions include the following four aspects. Firstly, new sufficient conditions for the existence of observer and asymptotical stability are developed by using an LMI approach, which can be easily solved via standard numerical software. Then, the design scheme presented can cope with the situation where the one-sided Lipschitz constant is unknown, making the complexity of this algorithm reduced significantly compared to almost all existing results. Thirdly, we remove the constraint of quadratic inner-boundedness which is a limitation imposed extensively by the existing works. Finally, by integrating an adaptation law the conservativeness is dropped sharply, which makes the applicable systems larger. In the end, the effectiveness and less conservativeness of results are tested in a series of numerical examples.

Stability analysis of uncertain delay differential equations

2021 ◽  
pp. 1-11
Author(s):  
Jian Wang ◽  
Yuanguo Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Liu Process ◽  
Uncertain Dynamical System ◽  
Analytic Expression ◽  
Functional Differential ◽  
The One ◽  
Delay Differential

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Unknown input fractional-order functional observer design for one-side Lipschitz time-delay fractional-order systems

2019 ◽  
Vol 41 (15) ◽  
pp. 4311-4321 ◽  
Author(s):  
Mai Viet Thuan ◽  
Dinh Cong Huong ◽  
Nguyen Huu Sau ◽  
Quan Thai Ha
Keyword(s):  
Time Delay ◽  
Nonlinear Systems ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Quadratic Function ◽  
Observer Design ◽  
Linear Matrix ◽  
Unknown Input ◽  
Functional Observer ◽  
The One

This paper addresses the problem of unknown input fractional-order functional state observer design for a class of fractional-order time-delay nonlinear systems. The nonlinearities consist of two parts where one part is assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition and the other is not necessary to be Lipschitz and can be regarded as an unknown input, making the wider class of considered nonlinear systems. By taking the advantages of recent results on Caputo fractional derivative of a quadratic function, we derive new sufficient conditions with the form of linear matrix inequalities (LMIs) to guarantee the asymptotic stability of the systems. Four examples are also provided to show the effectiveness and applicability of the proposed method.

scholarly journals Region of existence of multiple solutions for a class of Robin type four-point BVPs

2021 ◽  
Vol 41 (4) ◽  
pp. 571-600
Author(s):  
Amit K. Verma ◽  
Nazia Urus ◽  
Ravi P. Agarwal
Keyword(s):  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Source Term ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Monotone Iterative Technique ◽  
Nonlinear Boundary Value Problems ◽  
Existence Of A Solution ◽  
Monotone Iterative

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent.

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output

2018 ◽  
Vol 26 (3) ◽  
pp. 349-368 ◽  
Author(s):  
Alemdar Hasanov
Keyword(s):  
Parabolic Equation ◽  
Lipschitz Continuity ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Iteration Algorithm ◽  
Coefficient Problem ◽  
Inverse Coefficient Problem ◽  
Inverse Coefficient

AbstractThis paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional {J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}} corresponding to an inverse coefficient problem for the {1D} parabolic equation {u_{t}=(k(x)u_{x})_{x}} with the Neumann boundary conditions {-k(0)u_{x}(0,t)=g(t)} and {u_{x}(l,t)=0}. In addition, compactness and Lipschitz continuity of the input-output operator\Phi[k]:=u(x,t;k)\lvert_{x=0^{+}},\quad\Phi[\,\cdot\,]:\mathcal{K}\subset H^{1% }(0,l)\mapsto H^{1}(0,T),as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output {f(t):=u(0,t;k)} are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient. This lemma shows that an important advantage of gradient methods comes when dealing with the functionals of class {C^{1,1}(\mathcal{K})}. Specifically, this lemma asserts that if {J\in C^{1,1}(\mathcal{K})} and {\{k^{(n)}\}\subset\mathcal{K}} is the sequence of iterations obtained by the Landweber iteration algorithm {k^{(n+1)}=k^{(n)}+\omega_{n}J^{\prime}(k^{(n)})}, then for {\omega_{n}\in(0,2/L_{g})}, where {L_{g}>0} is the Lipschitz constant, the sequence {\{J(k^{(n)})\}} is monotonically decreasing and {\lim_{n\to\infty}\lVert J^{\prime}(k^{(n)})\rVert=0}.

scholarly journals On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

2020 ◽  
Vol 24 ◽  
pp. 39-55
Author(s):  
Julyan Arbel ◽  
Olivier Marchal ◽  
Hien D. Nguyen
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Random Variables ◽  
The Other ◽  
Bounded Support ◽  
Uniform Distributions ◽  
Standard Variance ◽  
Per Se ◽  
The One ◽  
The Relationship

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, Kumaraswamy, triangular, and uniform distributions.

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