There is a tendency of intensive development of a new scientific area aimed at optimizing the processes of comprehensive ensuring the life of society and industrial processes of countries, specifically logistics, and its more important aspect, military logistics. This paper considers typical contradictions between the need and opportunities for additional development of the theory of processes involving this system. On the one hand, the military has important, dynamic, multifaceted processes

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... the comprehensive provision of their combat operations to analyze, which requires significant intensification of the development of methods and models for quantitative analysis of the effectiveness of the functioning of military logistics systems. On the other hand, there is now limited availability of theoretical developments and the practical application of the necessary, convenient, effective mathematical tools aimed at computerization of solving the problems of providing military scientific and technical problems in real time. Matrix technology for forecasting the dynamics of functioning of closed systems of military logistics of various military purposes is proposed. Matrix calculus makes it possible to obtain intermediate and ultimate results in a compact form and carry out complex and cumbersome calculations using effective algorithms. A method to precisely solve the system of linear differential equations describing processes of arbitrary type has been proposed. The method is based on the use of the operational calculus by Laplace. The possibilities of the method and procedures of forecasting are illustrated by solving practical military tasks that arise during the functioning of military logistics systems of varying complexity. These tasks differ in configuration, different numbers of possible states, and state transitions