Generalized elements in problem on asymptotic attainability

We consider a problem which implies the choice of a solution subject to asymptotic constraints. We represent the results as ultrafilters of the space of ordinary estimates (the space is not necessarily endowed with a topology). This representation corresponds to an abstract attainability problem in its nonsequential asymptotic version.     

Nonsequential approximate solutions in abstract problems of attainability

The problem of constructing attraction sets in a topological space is considered in the case when the choice of the asymptotic version of the solution is subject to constraints in the form of a nonempty family of sets. Each of these sets must contain an “almost entire” solution (for example, all elements of the sequence, starting from some number, when solution-sequences are used). In the paper, problems of the structure of the attraction set are investigated. The dependence of attraction sets on the topology and the family determining “asymptotic” constraints is considered. Some issues concerned with the application of Stone-Čech compactification and the Wallman extension are investigated.

     

Generalized limits and representations of attraction sets in problems with constraints of asymptotic character

Doklady Mathematics -     

Extensions of abstract problems of attainability: Nonsequential version

Extension constructions of the problem of attainability in a topological space are studied. The constructions are based on compactification of the whole space of solutions or some of its fragments.     

Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the “complete” attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when there are no exact ordinary solutions, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactifier (an example is given showing that the similar property may be absent for other variants of the extension of the original problem).     

Attraction sets in abstract attainability problems: Equivalent representations and basic properties

We consider attainability problems in topological spaces under asymptotic constraints.     

Characterization of minimal elements in minimization problems with constraints

LetX be a compact Hausdorff space andC(X) be the set of all continuous functions defined onX. LetVC(X), and consider the problem of minimizing sup _xX W[x,v(x)], withvV. The functionW is a generalized weight function and can be chosen such that certain constraints are included.The notions of critical point and extremal signature are used to formulate characterization theorems for a minimal element inV. It is shown that these theorems hold only under certain conditions ofV andW. The results obtained are applied to the problem of the Chebyshev approximation with constraints and to the problem of optimization with strictly quasiconvex constraints.The work of the second author was supported in part by the Alexander von Humboldt Stiftung and the DAAD.     

Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature

A reachability problem with constraints of asymptotic nature is considered in a topological space. The properties of a rather general procedure that defines an extension of the problem are studied. In particular, we specify a rule that transforms an arbitrary extension scheme (a compactifier) into a similar scheme with the property that the continuous extension of the objective operator of the reachability problem is homeomorphic. We show how to use this rule in the case when the extension is realized in the ultrafilter space of a broadly understood measurable space. This version is then made more specific for the case of an objective operator defined on a nondegenerate interval of the real line.     

Existence results and asymptotic behavior for nonlocal abstract Cauchy problems

The purpose of this paper is to study the existence and asymptotic behavior of solutions for Cauchy problems with nonlocal initial datum generated by accretive operators in Banach spaces.     

On the question of construction of an attraction set under constraints of asymptotic nature

In a problem on the approximation of a vector function continuous on an interval by linear functions in the Chebyshev metric, necessary and sufficient conditions on the best approximation function are established.     

Error bounds for attainability sets of control systems with phase constraints

The problem of the error bounds for attainability sets of control systems described by ordinary differential equations under discretization of the phase constraints is studied. The peculiarity of the problems investigated in this paper is the phase constraints in the form of equality.     

Ultrafilters in the constructions of attraction sets: Problem of compliance to constraints of asymptotic character

We consider the properties of generalized elements in the problem of compliance to constraints of asymptotic character; these elements are identified with ultrafilters of special families of sets in the space of ordinary solutions.     

On the attainability problem under state constraints with piecewise smooth boundary

The paper is devoted to the problem of approximating reachable sets for a nonlinear control system with state constraints given as a solution set of a finite system of nonlinear inequalities. Each of these inequalities is given as a level set of a smooth function, but their intersection may have nonsmooth boundary. We study a procedure of eliminating the state constraints based on the introduction of an auxiliary system without constraints such that the right-hand sides of its equations depend on a small parameter. For state constraints with smooth boundary, it was shown earlier that the reachable set of the original system can be approximated in the Hausdorff metric by the reachable sets of the auxiliary control system as the small parameter tends to zero. In the present paper, these results are extended to the considered class of systems with piecewise smooth boundary of the state constraints.     

Extension of an abstract attainability problem with the use of the stone representation space

We consider an attainability problem in a complete metric space on values of an objective operator h. We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a nonempty family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.     

Partial asymptotic stability of abstract differential equations

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovskii theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 629–637, May, 2006.     

Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

Variational problems with constraints

Summary The problem considered is that of maximizing dt subject to x=G(x, y), x(0)=c, and0 ≤y≤x. An essentially new feature is determining in what regions y=x,0<y<x, and y=0.     

Stochastic control with exit time and constraints,application to small time attainability of sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed set K, i.e., the property that, for all arbitrary small time horizon T and for all initial condition in a sufficiently small neighborhood of K, there exists a solution to the controlled stochastic differential equation which reaches K before T.