Questions related to the extension of reachability problems and aimed at the construction of attraction sets, which are asymptotic analogs of reachable sets in the situation of successive relaxation of the constraint system, are studied. Finitely additive measures with the property of weak absolute continuity with respect to a fixed measure are used as generalized elements; the measure (in the case of control problems) is usually defined as the restriction of the Lebesgue measure to some family of measurable sets. The properties of relaxed reachability problems and the connection of their extensions with attraction sets in the class of ordinary solutions (controls), as well as the properties of these sets that have the sense of stability when the constraints are relaxed and asymptotic insensitivity when some “part” of the constraints is relaxed, are studied.

In control theory, the problem of constructing and investigating attainability domains is very important. However, under perturbations of constraints, this problem lacks stability. It is useful to single out the case when the constraints are relaxed. In this case, greater opportunities arise in terms of attainability, and often a useful effect can be observed even under slight relaxation of the constraints. This situation is analogous to the duality gap in convex programming. Very often, it is not possible to specify in advance how much relaxation of the constraints will occur. Therefore, attention is focused on the limit of the attainability domains under unrestricted tightening of the relaxed conditions. As a result, a certain attainability problem with asymptotic-type constraints arises. This problem formulation can be significantly generalized. Namely, we do not consider any unperturbed conditions at all and instead pose asymptotic-type constraints directly by means of a nonempty family of sets in the space of ordinary controls. Moreover, not only the case of control problems can be considered. In this general formulation, an analogue of the limit of attainability domains naturally appears as the relaxed conditions are infinitely tightened. For asymptotic constraints of this kind, we introduce solutions which are, at the conceptual level, similar to the approximate solutions of J.Warga, but we use filters or directedness, and not just sequences of ordinary solutions (controls). We investigate the most general attainability problem, in which asymptotic-type constraints can be generated by any nonempty family of sets in the ordinary solution space. It is shown, however, that the most practically interesting case is realized by filters, and the role of ultrafilters is noted as well. The action of constraints is associated with sets and elements of attraction. Furthermore, some properties of the family of all attraction sets are investigated.

Issues related to solving the additive problem of sequential traversal of sets with precedence restrictions and cost functions that allow dependence on the list of tasks are considered. The basic method is a broadly understood dynamic programming (DP), supplemented in the case of problems of appreciable dimension by decompositions of the family of tasks and transformation of the parameters of the original problem. Possible applications are related, in particular, to the problem of tool control in figured sheet cutting of parts on CNC machines. In this problem, an important circumstance is taking into account the precedence conditions, which have, in particular, the following meaning: in the case of a part with holes, cutting of each of the internal contours (corresponding to the holes) should precede cutting of the external contour. The quality criterion itself in this problem, as a rule, is additive. Another type of constraints concerns avoiding thermal deformations of parts. When using the approach with penalties for violating the conditions associated with effective heat dissipation during cutting, cost functions arise that allow dependence on the list of tasks completed to date. Note that in another applied problem, namely, in the problem of dismantling radiation hazardous objects, cost functions arise with dependence on the list of tasks that have not been completed at the moment (and, consequently, concern the objects that have not been dismantled). As a result, we arrive at a very general problem with precedence constraints and cost functions with dependence on the list of tasks. The decomposition applied in the case of a noticeable dimensionality with subsequent implementation of the DP requires, on the one hand, the development of clustering methods, and, on the other, the construction of an adequate structure for distributing global precedence conditions among clusters. In the theoretical part of the work, the case of two clusters is discussed, which makes it possible to cover with a single scheme a number of practically interesting problems of a range (in terms of dimensionality) type. An algorithm for constructing a composite solution is indicated, including a stage of clustering training based on a greedy algorithm. This “composite” algorithm is implemented on a PC; a computational experiment was carried out.

For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of relaxed controls is studied. The issues of properness of such a relaxation are considered; i.e., the possibility of approximating relaxed controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the ∗ -weak topology in the space of measures is established. The density of embeddings of ordinary controls and control–disturbance pairs in sets of corresponding relaxed controls in the ∗ -weak topologies is also shown.

We consider the problem of organizing a system of movement between the given points (cities) under conditions of resource constraints and in the presence of precedence conditions. The solvability conditions for this problem are extracted from the solution to the minimax traveling salesman problem (the bottleneck problem) without resource constraints. The solution to this extreme routing problem is determined on the basis of a broadly understood dynamic programming in its "non-additive" version. Possible applications may be related to the formation of the route of a vehicle (airplane or helicopter) in order to organize a transportation system in conditions of fuel shortage; it is assumed that in addition to the mandatory visits to all points, there are requirements for the passing movement of goods between some of the points, which creates additional restrictions (precedence conditions). To solve an auxiliary extremal problem, an optimal algorithm is constructed and implemented on a PC.

The problem of reachability in a topological space is studied under constraints of asymptotic nature arising from weakening the requirement that the image of a solution belong to a given set. The attraction set that arises in this case in the topological space is a regularization of certain kind for the image of the preimage of the mentioned set (the image and the preimage are defined for generally different mappings). When constructing natural compact extensions of the reachability problem with constraints of asymptotic nature generated by a family of neighborhoods of a fixed set, the case was studied earlier where the topological space in which the results of one or another choice of solution are realized satisfies the axiom T_2 . In the present paper, for a number of statements related to compact extensions, it is possible to use for this purpose a T_1 space, which seems to be quite important from a theoretical point of view, since it is possible to find out the exact role of the axiom T_2 in questions related to correct extensions of reachability problems. We study extension models using ultrafilters of a broadly understood measurable space with detailing of the main elements in the case of a reachability problem in the space of functionals with the topology of a Tychonoff power of the real line with the usual |·| -topology. The general constructions of extension models are illustrated by an example of a nonlinear control problem with state constraints.

This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin. In particular, this condition may arise in the tool control problem for thermal cutting machines with computer numerical control (CNC): if there are long parts among workpieces, the cutting process near a narrow material boundary should start with these workpieces since such parts are subject to thermal deformations, which may potentially cause rejects. The problem statement under consideration involves two zones for part processing. The aggregate routing process in the original problem includes a starting point, a route (a permutation of indices), and a particular track consistent with the route and the starting point. Each of the subproblems has specific precedence conditions, and the travel cost functions forming the additive criterion may depend on the list of pending tasks. A special two-stage procedure is introduced to apply dynamic programming as a solution method. The structure of the optimal solution is established and an algorithm based on this structure is developed. The algorithm is implemented on a personal computer and a computational experiment is carried out.

Ultrafilters of broadly understood measurable spaces and their application as generalized elements in abstract reachability problems with constraints of asymptotic nature are considered. Constructions for the immersion of conventional solutions, which are points of a fixed set, into the space of ultrafilters and representations of “limit” ultrafilters realized with topologies of Wallman and Stone types are studied. The structure of the attraction set is established using constraints of asymptotic nature in the form of a nonempty family of sets in the space of ordinary solutions. The questions of implementation up to any preselected neighborhood of the attraction sets in the topologies of Wallman and Stone types are studied. Some analogs of the mentioned properties are considered for the space of maximal linked systems.

This paper considers an optimal movement routing problem with constraints. One such constraint is due to decomposing the original problem into a preliminary subproblem and a final subproblem; the tasks related to the preliminary problem must be executed before the tasks of the final subproblem begin. In particular, this condition may arise in the tool control problem for thermal cutting machines with computer numerical control (CNC): if there are long parts among workpieces, the cutting process near a narrow material boundary should start with these workpieces since such parts are subject to thermal deformations, which may potentially cause rejects. The problem statement under consideration involves two zones for part processing. The aggregate routing process in the original problem includes a starting point, a route (a permutation of indices), and a particular track consistent with the route and the starting point. Each of the subproblems has specific precedence conditions, and the travel cost functions forming the additive criterion may depend on the list of pending tasks. A special two-stage procedure is introduced to apply dynamic programming as a solution method. The structure of the optimal solution is established and an algorithm based on this structure is developed. The algorithm is implemented on a personal computer and a computational experiment is carried out.

We consider a minimax routing problem related to visiting megacities under precedence conditions and cost functions with task list dependence. It is supposed that some megacity system requiring visiting above all is selected. For solving, an approach with decomposition into a set of two minimax routing problems is proposed. A two-step widely understood dynamic programming procedure realizing an optimal composition solution is constructed. The above-mentioned optimality is established by theoretical methods. Application of the results obtained is possible under investigation of multi-stage processes connected with regular allocation of resources. Another variant of application concerns the particular case of one-element megacities (i.e., cities) and may be related to the issues of aviation logistics under organization of flights using one tool (airplane or helicopter) under system of tasks on the realization of passing cargo transportation with prioritization of visits realized above all.

It is considered the routing problem for which some fixed tasks must be serviced above all. Other tasks can be serviced only after realization of above-mentioned original tasks. It is supposed that each our task is the megalopolis (nonempty finite set) visiting with fulfilment of some works. In our setting, two partial interconnected routing problems arise. We suppose that, in each partial routing problem, the corresponding precedence conditions are given. Using widely understood dynamic programming (DP), we obtain the optimal composition solution for initial total problem. As an application, we note the known engineering problem connected with sheet cutting by zones on CNC machines. By DP procedure the optimal algorithm realized on PC was constructed.

For a minimax routing problem with precedence conditions and cost functions that allow dependence on the list of tasks, we study the statement for which some of the tasks are allocated as first-priority ones. Other tasks can be started only after the fulfillment of priority tasks. The tasks themselves are connected with visiting megacities and, in particular, individual cities (terms corresponding to works in the field of solving the traveling salesman problem). One needs to find the extremum of arising two-stage problem with a minimax criterion, as well as the optimal compositional solution. In the paper, the optimal algorithm implemented on a PC is substantiated and built; a computational experiment is carried out. Possible applications may be related to some problems of aviation logistics in which it is required to ensure the visit of one vehicle (airplane or helicopter) to a system of aerodromes under a limited fuel reserve at each stage of the flight task; refueling is expected at points of visit (it is also assumed that a set of priority tasks is allocated).

DOI: 10.21538/0134-4889-2022-28-4-9-16 Поступила 9.09.2022 Принята к публикации 9.09.2022 Акопян Роман Размикович д-р. физ.-мат. наук, доцент, зав. отд. Институт математики и механики имени Н.Н. Красовского УрО РАН г. Екатеринбург e-mail: RRAkopyan@mephi.ru Антонов Николай Юрьевич д-р физ.-мат. наук, зам. директора Институт математики и механики имени Н.Н. Красовского УрО РАН

We consider a problem of sequential visiting of megalopolises under the preceding conditions and costs functions depending on the list of tasks currently unfulfilled. Selection of a routing process involving index permutation, trajectory and starting point is optimized; point of finish is optimized also. We use additive criterion consisting in summary costs of external (as for megalopolises) movings, costs of works related to visiting of megalopolises and assessments of the terminal state. Procedure of construction of optimal solution based on widely understood dynamic programming is investigated. The statement is focused on the problem of dismantling the system of radiation-hazardous sources; at the same time, it is assumed that not all sources are dismantled (it is possible when workers receive maximum doses of radiation), which requires evacuation in conditions of radiation exposure of sources that remain undismantled. A specific variant of the criterion is reduced to the summary dose of radiation received by an employee both at the stage of dismantling and at the stage of evacuation. An algorithm based on the theoretical constructions is proposed and realized on personal computer; a computational experiment is completed.

A constrained routing problem with complicated cost functions is studied. The construction of the cost functions can be difficult, and therefore the stages of this construction are elements of the solution of the problem. This situation arises, in particular, in studying the engineering problem of dismantling radiation hazardous elements, where, in the framework of a problem statement traditional for discrete optimization, it takes an unacceptably long time to construct a cost matrix whose entries characterize the radiation doses received by performers at the stage of displacement and dismantling. It is assumed that, at the stage of the computational implementation of the resulting optimal algorithm, the corresponding "parts" of the matrix may be not fed to the computer's memory but calculated as needed. Possible applications of the developed methods may be related to the problem of dismantling a decommissioned generator unit of a nuclear power plant.