Sufficient Conditions For Regularity Research Articles

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.

Switching law design for finite‐time stability of singular fractional‐order systems with delay

In this study, the authors present an analytical approach based on the Laplace transform and ‘inf-sup’ method for studying the finite-time stability of singular fractional-order switched systems with delay. A constructive geometric design for switching laws based on the construction of a partition of the stability state regions in convex cones is proposed. Using the proposed method, new delay-dependent sufficient conditions for regularity, impulse-free and finite-time stability of the system are developed in terms of tractable matrix inequalities and Mittag–Leffler functions. An example is provided to illustrate the effectiveness of the proposed method.

Regularity for 3D Navier-Stokes Equations Based on Energy Distribution at Wavenumber Bands

We present some regularity criteria for the Leray-Hopf weak solutions to the Cauchy problem for 3D Navier-Stokes equations based on energy distribution at wavenumber bands. We show sufficient conditions for regularity based on the ratios of energy dissipation rates at sufficiently high wavenumber bands and neighboring medium wavenumber bands. Moreover, we give a regularity criterion based on relations between energy at high and low wavenumber bands.

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, ut+uxxxx+∂xxux2=0, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition ux∈Lq′Lq(Q) is satisfied, where q,q′∈[1,∞] are such that either 1/q+4/q′<1 or 1/q+4/q′=1, q′<∞.

On the Stability of the Directional Regularity

In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.

The Diederich-Fornaess index and good vector fields

We consider the relationship between two sufficient conditions for regularity of the Bergman Projection on smooth, bounded, pseudoconvex domains. We show that if the set of infinite type points is reasonably well-behaved, then the existence of a family of good vector fields in the sense of Boas and Straube implies that the Diederich-Fornaess Index of the domain is equal to one.

Control of Fluctuations and Heavy Tails for Heat Variation in the Two-Time Measurement Framework

We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.

Rank one interval enclosure of the parametric united solution set

Consider linear algebraic systems $$A(p)x=b(p)$$ , where the matrix and the right-hand side vector depend linearly on a number of parameters $$p=(p_1,\ldots , p_K)^{\top }$$ that vary within given intervals. For each interval parameter, the structure of the dependencies can be presented by a finite sum of rank one matrices. This representation implies new more general and powerful sufficient conditions for regularity of a parametric interval matrix and a flexible general methodology for solving parametric interval linear systems with an expanded scope of applicability.

Necessary and sufficient conditions for quasi-strongly regularity of Graph Product

A $k$-regular graph ($k\ge 1$) with $n$ vertices is called a quasi-strongly regular graph with parameter $\lambda$ ($\lambda\in\NN$) if any two adjacent vertices have exactly $\lambda$ neighbors in common.A graph product is a binary operation on graphs. In this paper we prove some necessary and sufficient conditions for Decartes Product, Tensor Product, Lexicographical product and Strong product to be quasi-strongly regular.

Conservation of energy for the Euler\u2013Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.

On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations

This paper focuses on nonsmooth Newton methods of optimal control problems governed by mixed control–state constraints with differential algebraic equations. In contrast to previous results, we analyze lifting operators involved in nonsmooth Newton methods and establish corresponding convergence results. We also give sufficient conditions for regularity of generalized derivatives of systems of nonsmooth operator equations associated with optimal control problems.

Regularity of a Boundary Point for the p(x)-Laplacian

We study the behavior of solutions to the Dirichlet problem for the p(x)-Laplacian with a continuous boundary function. We prove the existence of a weak solution under the assumption that p is separated from 1 and ∞. We present a necessary and sufficient Wiener type condition for regularity of a boundary point provided that the exponent p has the logarithmic modulus of continuity at this point.

On the Truth, and Limits, of a Full Equivalence $${\mathbf{p \cong \,v^2 }}$$ p ≅ v 2 in the Regularity Theory of the Navier–Stokes Equations: A Point of View

The motivation at the origin of this note is the well known sufficient condition for regularity of solutions to the evolution Navier–Stokes equations, sometimes referred to in the literature as Ladyzhenskaya–Prodi–Serrin’s condition. Such a condition requires that the velocity field \(\,v\,\), alone, satisfies sufficiently strong integrability requirements in space–time. On the other hand, a relation like \(\,{p \cong \,|v|^2}\,\), with p pressure field, is loosely suggested by the Navier–Stokes equations themselves. In three papers published nearly 20 years ago we have considered this problem. The results obtained there immediately suggest new interesting questions. In this paper, we propose, and solve, some of them, while many other related problems remain still open.

Metrically Regular Differential Generalized Equations

In this paper we consider a control system coupled with a generalized equation, which we call a differential generalized equation (DGE). This model covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We study metric regularity and strong metric regularity of mappings associated with DGE by focusing in particular on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts. Metric regularity of a control system subject to inequality state-control constraints is characterized. A sufficient condition for local controllability of a nonlinear system is obtained via metric regularity. Sufficient conditions for strong metric regularity in function spaces are presented in terms of uniform pointwise strong metric regularity. A characterization of the Lipschitz continuity of the control part of the solution mapping as a function of time is established. Finally, a path-following procedure for a discretized DGE is proposed for which an error estimate is derived.

Minimization solutions to conservation laws with non-smooth and non-strictly convex flux

Conservation laws are usually studied in the context of sufficient regularity conditions imposed on the flux function, usually $C^{2}$ and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully extend the results. One specific nonlinear case is that of a piecewise linear flux function, for which we prove existence and uniqueness results. We also approximate it by a smoothed, superlinearized version parameterized by $\varepsilon$ and consider the characterization of the minimizers for the smooth version and limiting behavior as $\varepsilon\downarrow0$ to that of the sharp, polygonal problem. In proving a key result for the solution in terms of the value of the initial condition, we provide a stepping stone to analyzing the system under stochastic processes, which will be explored further in a future paper.

Ramsey properties of nonlinear Diophantine equations

We present general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework.Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters βN, grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers N⁎.

Sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets

In this paper, two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices. In the first approach, a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained. In the second approach, a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions. As an application, the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices. Two algorithms are proposed and results obtained are compared with those obtained by earlier methods and Monte Carlo simulations. The advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets. The second algorithm is very effective compared to the first algorithm from the computational time point of view. Several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work.

Enclosing the solution set of parametric interval matrix equation A(p)X = B(p)

Consider the parametric matrix equation A(p)X = B(p), where the elements of the matrices A(p) and B(p) depend linearly on a number of uncertain parameters varying within given intervals. We prove that the united parametric solution sets of the matrix equation and that of the corresponding linear system with multiple right-hand sides, although different as sets, have the same interval hull. A generalization of the parametric Krawczyk iteration with low computational complexity for the matrix equation is presented. Some details improving the implementation and the application of this method are discussed. An interval method, designed by A. Neumaier and A. Pownuk for enclosing the united solution set of parametric linear systems with particular dependency structure, is generalized for arbitrary linear dependencies between the parameters and for systems with multiple right-hand sides. A new, more powerful, sufficient condition for regularity of a parametric interval matrix is proven. An important application of the linear systems with multiple right-hand sides is presented as a key methodology for feasibility in computing the interval hull of a class of united parametric solution sets that appear in practical problems.

Admissibility analysis for discrete-time singular Markov jump systems with asynchronous switching

This paper deals with admissibility analysis for discrete-time singular Markov jump systems (SMJSs) with asynchronous switching, which means the switching of controllers is asynchronous with the switching of subsystems. The switching delay and state delay are modeled as Markov chains. The resulting closed-loop system is modeled as a singular system with a Markovian jumping vector and two Markovian jumping parameters. A new necessary and sufficient condition for regularity, causality and stability of the closed-loop system is proposed. Based on this, a state feedback controller is designed in terms of a strict linear matrix inequality (LMI) to guarantee the admissibility of SMJSs with asynchronous switching. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Harmonic interpolation of Hermite type based on Radon projections in two directions

We study Hermite-type interpolation based on Radon projections. We give necessary and sufficient conditions for regularity in the cases in which chords lie in two directions. We also prove that the interpolation polynomials are continuous with respect to the distance from the origin to the chords.