A. G. Chentsov

Dynamic Programming in the Routing Problem with Constraints and Costs Depending on a List of Tasks

637 We study the problem of visiting cities with ante� cedence conditions and traveling costs depending on a list of tasks. The formulation of the problem is associ� ated with the dismantling of a shut down nuclear power unit and is a generalization of that considered in (1). Conceptually, the problem under study goes back to the wellknown traveling salesman problem (2-4), but differs from the latter in having several important features. The solution method used follows (5-7) but admits a higher degree of generality in defining the cost functions. Specifically, in contrast to (1), it is assumed that the arrival and departure points in visited cities can be different, which is important for the abovementioned application. As was noted, the cost functions (traveling and internal operation) can depend on the list of tasks that have not been per� formed by the current moment. This kind of depen� dence was not considered in (5-7). This work deals with qualitative issues. A version of the dynamic programming method (DPM) is pro� posed that takes into account the above features (including the antecedence constraints) and is imple� mented without constructing the entire array of Bell� man function values. Additionally, we suggest an iter� ation method based on an equivalent transformation of the original problem to a simpler form.

Route problem with constraints depending on a list of tasks

An additive route problem with preceding conditions is considered in which the cost function and the move constraints both depend on a list of tasks that have not been performed by the current time. The problem is solved by applying a dynamic programming method that takes into account both these factors and is implemented in the construction of a (generally) incomplete array of Bellman function values.

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

We study a variant of the reachability problem with constraints of asymptotic character on the choice of controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situation is complicated by the presence of discontinuous dependences, which produce effects of the type of multiplying a discontinuous function by a generalized function. The constructed extensions in the special class of finitely additive measures make it possible to present the required solution, defined as an asymptotic analog of a reachable set, in terms of a continuous image of a compact, which is described with the use of the Stone space corresponding to the natural algebra of sets of the control interval.One of the authors had the honor of communicating with Nikolai Nikolaevich Krasovskii for many years and discussed with him problems that led to the statement considered in the paper. Krasovskii’s support of this research direction provided possibilities for its fruitful development. His disciples and colleagues will always cherish the memory of Nikolai Nikolaevich in their hearts.

Programmed iteration method in packages of spaces

An abstract version of the programmed iteration method (used in the theory of differential games) is considered that is intended for the problem of observing phase constraints on a nonempty subset of the real line. It is assumed that trajectories and uncontrolled disturbances are realized in packages of mutually inconsistent spaces. This consistency is postulated to investigate solvability conditions for the problem in the class of quasi-strategies. As a result, the set of successful solvability is constructed and the structure of resolving strategies is determined.

Programmed iteration method and sets of positional absorption

A pursuit–evasion differential game is considered, and the programmed iteration method is used to construct a set of positional absorption corresponding to the Krasovskii–Subbotin alternative theorem. The case is considered where the set of positions determining the state constraints may not be closed (in the position space), but has closed sections corresponding to fixed times. Properties are established that are interpreted as the (one-sided) continuity of the positional absorption set from above, and the relation to the solution of the game in the class of set-valued quasi-strategies is shown.

On an asymptotic analysis problem related to the construction of an attainability domain

Problems of constructing and analyzing the properties of attainability domains play an important role in control theory and its applications. In particular, this applies to control under impulse constraints that reflect the energetics of a process. The situation is complicated by the possible instability of the process under variation (in particular, under relaxation) of constraints related to boundary and intermediate conditions. Stability of the problem is also missing, in general, under relaxation of state constraints. In these cases, it is natural to focus on the asymptotic variant of the statement; this is especially expedient when one has to deal with initially asymptotic requirements. In all these cases, it seems expedient to use analogs of J. Warga’s approximate solutions. At the same time, to seek necessary approximate (and, in fact, asymptotic) solutions, it is natural to use generalized modes. For problems with impulse constraints and discontinuity in the coefficients of control actions, such modes lead to phenomena described by products of discontinuous functions and generalized functions even in the class of linear systems. In a large series of his studies, to overcome the arising difficulties, one of the authors used constructions of extension in the class of finitely additive measures. The present paper follows this approach and is ideologically relevant to the engineering problem of controlling the thrust of an engine under conditions of a given program of variation of its orientation; it is postulated that energy resources are completely consumed in a natural (for a number of impulse control problems) mode of short-duration impulses: the set of time instants at which the instantaneous control is different from zero can be embedded in an interval of vanishingly small length. Within these short periods of time, the engine should consume all energy resources while obeying some other constraints (making the sense of moment constraints) to a high degree of accuracy. In addition, one should take into account the possible discontinuity of the functions defining the coefficients of control actions. As a natural analog of the attainability domain, we use an attraction set, whose construction, together with the subsequent study of its main properties, constitutes the goal of the present study.

On a Generalization of One Game Control Problem in the Class of Finitely Additive Measures

We consider a terminal game control problem for a linear system with discontinuous control coefficients subject to impulse constraints. We construct a generalized game control problem in the class of finitely additive measures with the property of the weak absolute continuity with respect to the restriction of the Lebesgue measure to some “sufficient” measurable structure.