The routing problems with optimization of the starting point: dynamic programming

The extreme routing problem focused on engineering applications in mechanical engineering is considered. We mean the well-known task of tool controlling in the CNC sheet cutting machines. A mathematical model is presented which includes a system of megalopolises (nonempty finite sets) and cost functions depending on the list of tasks. Megalopolises are constructed on the basis of discretization of equidistant curves of part contours. The dependence on the list of tasks is connected with reasons associated with the dynamic constraints that arise in the process of task completion. Among all restrictions, the conditions of precedence are distinguished (earlier cutting of the inner contours and more earlier cutting of large parts). Rational consideration of the precedence conditions allows one to reduce the complexity of calculations when widely understood dynamic programming (DP) is used in the implementation that develops R. Bellman's scheme. This approach makes it possible to solve the problem of optimizing complexes, which include the initial state (starting point), the method of numbering megalopolises in the order of their visits, and the specific trajectory of the process. For a problem complicated by the dependence of the terminal function on the initial state, a decomposition algorithm is used, which allows, in a substantial part of the procedure, the application of a single (for all initial states) DP scheme. The optimal algorithm based on DP is implemented as a program for PC; a computational experiment is conducted

In memory of Arkady Viktorovich Kryazhimskiy (1949-2014)

The article is devoted to the description of Academician Arkady Kryazhimskiy's life path. The facts of the scientific biography of Acad. Kryazhimskiy are presented with the emphasis on his outstanding contribution into the theory of dynamic inversion, the theory of differential games, and control theory. His personal talents in different spheres are also marked out

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 choiceof controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situaton is complicated by the presence of discontinuou dependences, which produce effects of the type of multiplying a discontinuous function by a generalizd 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 orresponding 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 nd discussed with him problems that led to the statement considered in the paper. Krasovskii.s support of this research direction provided possbilities for its fruitful development. Hi

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 energtics of a process. The situation is complicated by the possible instability of the process under variation (in particular, under relaxatin) 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 problem 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 overcme 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 conditons 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 reources 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