Reconstruction of noisy signals by minimization of non-convex functionals

Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals.     

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.     

On the pseudo-monotonicity of generalized gradients of nonconvex functions

The paper is devoted to study the class of nonconvex, nondifferentiable functionals on a reflexive Banach space whose generalized Clarke's gradient i s pseudo-monotone i n the sense of Browder-Hess. I n particular, it has been proved that on some restrictions functionals expressed as a pointwise minimum of a finite collection of convex functions belong to this class. Results obtained are used to establish some existence theorems for hemivariational inequalities involving superpotentials under consideration.     

On converse Lyapunov theorems for fluid network models

We consider the class of closed generic fluid network (GFN) models, which provides an abstract framework containing a wide variety of fluid networks. Within this framework a Lyapunov method for stability of GFN models was proposed by Ye and Chen. They proved that stability of a GFN model is equivalent to the existence of a functional on the set of paths that is decaying along paths. This result falls short of a converse Lyapunov theorem in that no state-dependent Lyapunov function is constructed. In this paper we construct state-dependent Lyapunov functions in contrast to path-wise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem. To resolve this problem we introduce the class of strict GFN models by forcing closed GFN models to satisfy a concatenation and a semicontinuity condition. For the class of strict GFN models we define a state-dependent Lyapunov function and show that a converse Lyapunov theorem holds. Finally, it is shown that common fluid network models, like general work-conserving and priority fluid network models as well as certain linear Skorokhod problems define strict GFN models.     

Intervallfunktionale vom GAUSS-LEGENDRE-Typ

A general class of quadrature formulas is discussed where the usual point functionals are replaced by interval functionals. The quadrature formulas of this type with highest possible degree of exactness are minimizing a certain functional. This leads to an existence theorem for arbitrary weights, which must comply with some necessary condition only. Besides, this minimum property enables us to calculate the nodes of such quadrature formulas very effectively.     

A nonsmooth approach to nonexpected utility theory under risk

We consider concave and Lipschitz continuous preference functionals over monetary lotteries. We show that they possess an envelope representation, as the minimum of a bounded family of continuous vN-M preference functionals. This allows us to use an envelope theorem to show that results from local utility analysis still hold in our setting, without any further differentiability assumptions on the preference functionals. Finally, we provide an axiomatisation of a class of concave preference functionals that are Lipschitz.     

A nonsmooth approach to nonexpected utility theory under risk

We consider concave and Lipschitz continuous preference functionals over monetary lotteries. We show that they possess an envelope representation, as the minimum of a bounded family of continuous vN-M preference functionals. This allows us to use an envelope theorem to show that results from local utility analysis still hold in our setting, without any further differentiability assumptions on the preference functionals. Finally, we provide an axiomatisation of a class of concave preference functionals that are Lipschitz.     

Lagrange approach to the optimal control of diffusions

A new approach to the optimal control of diffusion processes based on Lagrange functionals is presented. The method is conceptually and technically simpler than existing ones. A first class of functionals allows to obtain optimality conditions without any resort to stochastic calculus and functional analysis. A second class, which requires Ito's rule, allows to establish optimality in a larger class of problems. Calculations in these two methods are sometimes akin to those in minimum principles and in dynamic programming, but the thinking behind them is new. A few examples are worked out to illustrate the power and simplicity of this approach.Research performed at the Mathematisches Seminar der Universität Kiel with support provided by an Alexander von Humboldt Foundation fellowship.     

Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands

It is known that sequential weak lower semicontinuity and weak-strong convergence (in the scalar case) properties of integral functionals may be characterized by means of their integrands. In this paper we introduce a Young measure approach obtaining both these results and the characterization for the second property in the vector-valued case. We discuss also motivations for the definition of strict quasiconvexity, and point out that the characterization of the classes of functionals having weak-strong convergence property everywhere is not a trivial problem in the general case.     

Existence of non-trivial solutions for systems of n fourth order partial differential equations

In this paper, employing a very recent local minimum theorem for differentiable functionals due to Bonanno, the existence of at least one nontrivial solution for a class of systems of n fourth order partial differential equations coupled with Navier boundary conditions is established.     

Optimality of Euler-integral information for solving a scalar autonomous ode

We study the minimum error of algorithms for solving a scalar autonomous ODE. Information is defined asn evaluations of functionals of the right hand sidef. These functionals can be linear or nonlinear and continuous. Euler-integral information is defined and its optimality is proved in a class of regularf.     

A class of functionals on a banach space for which strong and weak local minima do coincide ∗

We present a class of functionals on a real Banach space for which every local minimum point with respect to the norm topology is so also with respect to the weak topology     

BOUNDEDNESS IN p-NFDES

1 IntroductionRecently, the papers (cf [1], [2]) have established stabi1ity of p-NFDEsusing Liapunov-Razumikhin functionaJs. However, in order to study the limtset and the existence of periodic solutions of p-NFDEs, we need to know notonly the stability but ajso the boundedness of solutions. In this paper, weshal1 discuss the boundedness properties of P-NFDEs by Liapunov functionals,which improve some boundedness conditions for FDEs in the literature. Someexamples are also worked out t…     

Global exponential stability for a class of retarded functional differential equations with applications in neural networks

In this paper, the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time-varying delays are studied by employing several Lyapunov functionals. A number of sufficient conditions for these types of stability are presented. Our results show that these conditions are milder and more general than previously known criteria, and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Furthermore, the results obtained for neural networks with time-varying delays do not assume symmetry of the connection matrix.     

Local Stabilization of the Navier–Stokes Equations with a Feedback Controller Localized in an Open Subset of the Domain

We study the local exponential stabilization, near a given steady-state flow, of solutions of the 2-D and 3-D Navier–Stokes equations in a bounded domain. The control is effectuated through a distributed control localized in an open subset of the domain. We apply a linear feedback controller, obtained by the solution of a control problem. Lyapunov functionals are given for a large class of cost functionals.     

A regularity result for a minimum problem in orlicz-sobolev spaces with applications in the study of the dirichlet problem for the operator of hencky-nadai theory

The existence and uniqueness of the solution to the problem of minimum for functionals generated by N- functions are obtained in Orlicz-Sobolev spaces. Applications for some functionals dealing with Hencky theory are given.     

A constrained minimization problem with integrals on the entire space

In this paper we consider the question of minimizing functionals defined by improper integrals. Our approach is alternative to the method of concentration-compactness and it does not require the verification of strict subaddivity.Dedicated to the memory of Antonio Gilioli (1945–1989)     

Optimizing the smoothed bootstrap

In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.     

Some estimates of the minimizing properties of web functions

We study a general class of functionals over and we approximate their infimum by means of the minimum in the space of web functions. We provide several tools for estimating this approximation and we study in detail some meaningful models. Received: 13 July 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Generalized Willmore functionals and related variational problems

The purpose of this paper is to study the conformally invariant functionals of hypersurfaces in a Riemannian manifold and variational problems related to these functionals. A class of conformal invariants is presented and the variational problem related to this class of conformally invariant functionals is studied.