The Features of Solving of the set Partitioning Problems with Moving Boundaries Between Subsets

Summary Problems and methods presented in this paper synthesize foundations of theory of continuous set partitioning problems (SPP) and optimal control of systems described by ordinary differential equations. In order to mathematically formulate SPP quite often one should take into account the temporal and spatial changes of object or process state. Some of such models concerned with problems of preservation of the environment were learning by our scientists. Mathematical models of problems mentioned above are new from the viewpoint of problem statement and interesting to further generalization and developing of theoretical results which could be used in practice widely. A common SPP could be formulated as follows: it is necessary to partition a given area (set) into a finite number of disjoint subsets that satisfies certain restrictions so that the objective function reaches an extreme value. We propose a new problem statement, which differs from known ones in the following way: the desired set partition is dynamic in consequence of 1) a function which describes the certain object or process state varies with time; 2) a function, choice of which has an inuence on state of this object or process, is defined by partition of considered set each moment of time. This problem amounts to optimal control one for which one should write out the necessary conditions of optimality in the form of Pontrjagin‘s maximum principle. The constructed algorithm for solving such problems bases on combining both the methods of solving continuous SPP and methods of optimal control theory. With a view to investigate the properties of solutions of new set partitioning problem we realized the series of computational experiments and made qualitative analysis of obtained results.