On almost rc-Lindelöf sets

In [4], Dlaska introduced the class of almost rc-Lindelöf sets and studied some basic properties of such sets. In this paper, we obtain further results concerning almost rc-Lindelöf sets. We also introduce new concepts to obtain several mapping properties concerning almost rc-Lindelöf sets and almost rc-Lindelöf spaces. The property of being an almost rc-Lindelöf set is invariant under functions which are slightly continuous and weakly θ-irresolute. It is also shown that the property of being an almost rc-Lindelöf space is inverse invariant under functions which are weakly almost open, ω-regular open, and whose fibers are S-sets.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

对称连续分布函数的最优不变估计

该文考虑了未知对称连续分布函数的不变估计问题.连续分布函数在单调变换群下是不变的^[1], 但这个变换群不能保证对称分布函数的不变性.于是, 所要研究的判决问题在单调变换群下不再是不变的. 为了保证判决问题不变性, 考虑一个新的变换群—单调奇变换群, 它确保了所研究的判决问题的不变性.注意到对称分布函数零点的特殊性质, 即, 对任一对称分布函数F, 均有F(0)=1/2,通过视零点为一伪观察值, 得到了所有的非随机化不变估计, 并在不变估计中找到了最优不变估计.     

Strong convergence for maximal monotone operators, relatively quasi-nonexpansive mappings, variational inequalities and equilibrium problems

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function.     

Tracking within a time interval on the basis of data supplied by finite observers

We solve the tracking control problem, in which one should bring a trajectory of a system of linear ordinary differential equations into a neighborhood of a trajectory of another system within a given time interval. After getting into this neighborhood, one should keep the trajectory of the first subsystem in it for a time interval of given duration. For the control synthesis, we use incomplete and imprecise information on the online deviation of one trajectory from the other, which is obtained in real time from linear equations of observation. We consider distinct structures of observers, which substantially affect the solution of control problems for such systems. The equations of dynamics and admissible measurements contain uncertainty for which one knows only some hard pointwise constraints. To solve the main problem, we use an approach that can be reduced to the construction of auxiliary information sets and weakly invariant sets with a subsequent “aiming” of one set at a tube. We suggest an efficient method for an approximate solution on the basis of ellipsoidal calculus techniques. The results of the algorithm operation are illustrated by an example of the solution of a tracking control problem for two fourth-order subsystems.     

Bounds for a domain containing all compact invariant sets of the system describing the laser–plasma interaction

In this paper we consider the localization problem of compact invariant sets of the system describing the laser–plasma interaction. We establish that this system has an ellipsoidal localization for simple restrictions imposed on its parameters. Then we improve this localization by applying other localizing functions. In addition, we give sufficient conditions under which the origin is the unique compact invariant set.     

Dynamical systems on lattices with decaying interaction I: A functional analysis framework

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites.We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local.We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].     

Stabilization for hybrid stochastic systems by aperiodically intermittent control

In this paper, the mean-square exponential stabilization for stochastic differential equations with Markovian switching is studied. Specifically, a new set of sufficient conditions is derived to obtain the aperiodically intermittent control design which exponentially stabilizes the addressed hybrid stochastic differential equations. Further, stabilization problem by periodically intermittent control can be deduced as a special case from the developed results. As an application, we consider the Hopfield neutral network model with simulations to illustrate the effectiveness of developed aperiodically intermittent control design.     

On the strong CE-property of convex sets

We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.     

Extension of quasi-Lipschitz set functions

In this article we consider so-calledN -triangular and quasi-Lipschitz set functions. In terms ofN -semimeasures, we establish necessary and sufficient conditions for extending a quasi-Lipschitz set function which is continuous from above at zero from a ring of sets to the σ-ring generated by these sets, and also conditions for the uniqueness of the extension. As simple corollaries we obtain analogous results for vector-valued measures, continuous triangular measures, and real-valued finiteN -triangular set functions which are continuous from above at zero.     

Weakly convex sets and modulus of nonconvexity

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.     

Nonclassical problem with integral boundary conditions for elliptic system

In the present paper, a mixed nonclassical problem for multidimensional second-order elliptic system with Dirichlet and nonlocal integral boundary conditions is considered. Since Lax-Milgram theorem cannot be applied straightforwardly for such a nonlocal problem, we consider the problem in the spaces of vector-valued distributions with respect to one space variable with values in the spaces of functions with respect to the other space variables. We introduce special multipliers and applying them we obtain suitable new a priori estimates, and under minimal conditions on the coefficients of the elliptic operator we prove the existence and uniqueness of the solution in appropriate spaces of vector-valued distributions with values in Sobolev spaces.     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Some properties for the attractors

For the continuous flows defined on a topological space, we have discussed some properties for the invariant sets and their domains of influence. We have proved the following open problem posed by C. Conley: an attractor in a locally connected compact Hausdorff invariant set has finitely many components. In the meantime, two necessary and sufficient conditions for a set to be an attractor have been given.     

Depth-Optimized Convexity Cuts

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems.     

Free Interpolation in the Spaces of Analytic Functions With Derivative of Order <Emphasis Type="Italic">s</Emphasis> From the Hardy Space

We consider the problem of free interpolation for the spaces of analytic functions with derivative of order s in the Hardy space H^p. For the sets that satisfy the Stolz condition, we obtain a condition necessary for interpolation: if 1 ≤ p < ∞, then the set must be a union of s sparse sets. For p = ∞, we obtain a necessary and sufficient condition for interpolation: the set must be a union of s + 1 sparse sets. In this case, we construct an extension operator. Bibliography 11 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 169–202.     

Efficient and weakly efficient solutions for the vector topical optimization

In this paper, we consider some scalarization functions, which consist of the generalized min-type function, the so-called plus-Minkowski function and their convex combinations. We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space. Then, we establish some versions of Farkas type results for the infinite inequality system involving vector topical functions. As applications, we obtain the necessary and sufficient conditions of efficient solutions and weakly efficient solutions for a vector topical optimization problem, respectively.