Sufficient Conditions For Regularity Research Articles

REGULAR VECTOR-FIELDS IN BANACH SPACES

It is known that regular vector-fields generate convergent discrete and continuous descent methods for minimizing functions. In this note we present necessary and sufficient conditions for regularity of descent vectorfields associated with convex and Lipschitz functions defined on general Banach spaces.

Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces

In this paper we give new results on complete abstract second order differential equations of elliptic type in the framework of Hölder spaces, extending those given in [4] and [5]. More precisely we study $u^{\prime \prime }+2Bu^{\prime }+Au=f$ in the case when $f$ is Hölder continuous and under some natural assumptions on the operators $A$ and $B$. We give necessary and sufficient conditions of compatibility to obtain a strict solution $u$ and also to ensure that the strict solution has the maximal regularity property.

The asymptotics of the optimal value of the performance functional in a linear optimal control problem in the regular case

An optimal control problem is investigated for a linear system with fast and slow variables, a convex terminal performance functional depending on the slow variables, and smooth geometric constraints on the control. Sufficient regularity conditions are presented for the asymptotics of a solution of this problem, and a complete asymptotic expansion of the optimal value of the performance functional in powers of a small parameter is constructed.

Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

We investigate Holder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Holder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Holder continuous, while they have the two sided lower Holder continuity property for less regular constraints. Finally, we provide sufficient conditions for Holder type regularity of optimal controls.

Regularity and Recurrence of Switching Diffusions

This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are “linearizable” with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients (averaged with respect to the stationary measure of the Markov chain).

Minimum-Length Trajectories for a Car: An Example of the Use of Boltianskii's Sufficient Conditions for Optimality

We consider the kinematic model of a car which describes the rolling-without-slipping constraint of the wheels on an horizontal floor and the bound on the angle of steering of front wheels. The problem of determining shortest paths for such a vehicle is known as the Reeds and Shepp's problem. Ten years ago, a complete solution to this problem was determined on the basis of a complex reasoning grounded on the necessary conditions of Pontryagin's Maximum Principle and completed with a set of geometric arguments. In this note, we provide a simple new proof of the optimality of this construction by using a verification theorem based on Boltianskii's sufficient regularity conditions. To our knowledge, it is the first example of a regular synthesis for a nonholonomic system in a three-dimensional space

A Sufficient Condition of Regularity for Axially Symmetric Solutions to the Navier–Stokes Equations

In the present paper, we prove a sufficient condition of local regularity for suitable weak solutions to the Navier–Stokes equations having axial symmetry. Our condition is an axially symmetric analogue of the so-called $L_{3,\infty}$-case in the general local regularity theory.

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.

Linear regularity and ϕ-regularity of nonconvex sets

In this paper, we discuss some sufficient conditions for the linear regularity and bounded linear regularity (and their variations) of finitely many closed (not necessarily convex) sets in a normed vector space. The accompanying necessary conditions are also given in the setting of Asplund spaces.

Regular submodules of torsion modules over a discrete valuation domain

A submodule W of a p-primary module M of bounded order is known to be regular if W and M have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.

On the Conserved Quantities for the Weak Solutions of the Euler Equations and the Quasi-geostrophic Equations

In this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible Euler equations. The regularity of the weak solutions are measured in terms of the Triebel-Lizorkin type of norms, \(\dot {\mathcal F}^s_{p,q}\) and the Besov norms, \(\dot {\mathcal B}^s_{p,q}\). In particular, in the Besov space case, our results refine the previous ones due to Constantin-E-Titi (energy) and the author of this paper (helicity), where the regularity is measured by a special class of the Besov space norm \(\dot {\mathcal B}^s_{p,\infty}= \dot {\mathcal N}^s_p\), which is the Nikolskii space. We also obtain a sufficient regularity condition for the conservation of the Lp-norm of the temperature function in the weak solutions of the quasi-geostrophic equation.

Torsional barriers and nonlinear optical properties of 2-, 3-, 4-phenylpyridine molecules

A submodule $W$ of a p-primary module $M$ of bounded order is known to be regular if $W$ and $M$ have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.

Navier–Stokes Equations: Almost L3,∞-Case

A sufficient condition of regularity for solutions to the Navier–Stokes equations is proved. It generalizes the so-called L3,∞-case.

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations

We prove two sufficient conditions for local regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. One of these conditions implies the smoothness of L3,∞-solutions as a particular case. Bibliography: 12 titles.

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on ℝN, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.

On the regularity up to the boundary in the theory of the Navier–Stokes equations with generalized impermeability conditions

In this Note, we extend sufficient conditions for regularity we described in our previous works so that they are valid not only in the interior, but up to the boundary of a flow field. The conditions are based on the integrability properties of either one of the eigenvalues of the rate of deformation tensor or one component of velocity. To cite this article: J. Neustupa, P. Penel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Generalization of a Parametric Fixed‐Point Iteration

AbstractBased on new sufficient regularity conditions, Rump's method for solving parametric interval linear systems is generalized which expands its scope of applicability over a class of co‐called column‐dependent parametric matrices. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Maximum likelihood analysis of masked series system lifetime data

Component lifetime parameters of a series system are estimated from system lifetimes and masked cause of failure observations. The time and cause of system failures are assumed to follow a competing risks model. The masking probabilities of the minimum random subsets are not subjected to the symmetry assumption. Sufficient regularity conditions are provided, justifying the maximum likelihood analysis. Maximum likelihood estimates of both the lifetime parameters and masking probabilities are generically computed via an EM algorithm. An appropriate set of asymptotically pivotal quantities are also derived. Such maximum likelihood based estimates are further refined by bootstrap. The developed techniques are illustrated by numerical examples of independent Weibull component lifetimes with distinct scale and shape parameters.

Sufficient Conditions for Regularity and Strict Identifiability in MIMO Systems

We consider regularity and identifiability of convolutive multi-input multi-output (MIMO) systems with additive white Gaussian noise, modeling the source and finite impulse response (FIR) channel sequences as deterministic unknowns. In the blind context, the MIMO system is not locally identifiable; hence, its Fisher information matrix (FIM) is not regular. In fact, the dimension of the complex-valued blind FIM null space is at least the number of sources squared. Because the FIM is singular, additional information about the system is required to resolve the degrees of uncertainty and thereby obtain a valid Cramer-Rao bound (CRB); therefore, it is of interest to know under what conditions the blind FIM nullity reaches its lower bound. We develop sufficient conditions for the complex FIM to attain its minimum nullity, refining previous necessary conditions, and extending single-input multi-output (SIMO) results. We show that the sufficient conditions for the complex FIM to have minimum nullity are also equivalent to sufficient conditions for MIMO strict identifiability. These provide sufficient conditions on the richness of the sources, the required diversity, and the source lengths. Under these conditions, additional constraints, such as training, may be employed to yield an identifiable system with no ambiguities remaining.