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In this paper we consider mathematical model to control water quality. We study a system with two-level hierarchy: one environmental organization (supervisor) at the top level and a few industrial enterprises (agents) at the lower level. The main goal of the supervisor is to keep water pollution level below certain value, while enterprises pollute water, as a side effect of the manufacturing process. Supervisor achieves its goal by charging a penalty for enterprises. On the other hand, enterprises choose how much to purify their wastewater to maximize their income.The fee increases the budget of the supervisor. Moreover, effulent fees are charged for the quantity and/or quality of the discharged pollution. Unfortunately, in practice, such charges are ineffective due to the insufficient tax size. The article solves the problem of determining the optimal size of the charge for pollution discharge, which allows maintaining the quality of river water in the rear range.

We describe system members goals with target functionals, and describe water pollution level and enterprises state as system of ordinary differential equations. We consider the problem from both supervisor and enterprises sides. From agents’ point a normal-form game arises, where we search for Nash equilibrium and for the supervisor, we search for Stackelberg equilibrium. We propose numerical algorithms for finding both Nash and Stackelberg equilibrium. When we construct Nash equilibrium, we solve optimal control problem using Pontryagin’s maximum principle. We construct Hamilton’s function and solve corresponding system of partial differential equations with shooting method and finite difference method. Numerical calculations show that the low penalty for enterprises results in increasing pollution level, when relatively high penalty can result in enterprises bankruptcy. This leads to the problem of choosing optimal penalty, which requires considering problem from the supervisor point. In that case we use the method of qualitatively representative scenarios for supervisor and Pontryagin’s maximum principle for agents to find optimal control for the system. At last, we compute system consistency ratio and test algorithms for different data. The results show that a hierarchical control is required to provide system stability.

Modeling combat operations is an urgent scientific and practical task aimed at providing commanders and staffs with quantitative grounds for making decisions. The authors proposed the function of victory in combat and military operations, based on the function of the conflict by G. Tullock and taking into account the scale of combat (military) operations. On a sufficient volume of military statistics, the scale parameter was assessed and its values were found for the tactical, operational and strategic levels. The game-theoretic models «offensive – defense», in which the sides solve the immediate and subsequent tasks, having the formation of troops in one or several echelons, have been investigated. At the first stage of modeling, the solution of the immediate task is found — the breakthrough (holding) of defense points, at the second — the solution of the subsequent task — the defeat of the enemy in the depth of the defense (counterattack and restoration of defense). For the tactical level, using the Nash equilibrium, solutions were found for the closest problem (distribution of the forces of the sides by points of defense) in an antagonistic game according to three criteria: a) breakthrough of the weakest point, b) breakthrough of at least one point, and c) weighted average probability. It is shown that it is advisable for the attacking side to use the criterion of «breaking through at least one point», in which, all other things being equal, the maximum probability of breaking through the points of defense is ensured. At the second stage of modeling for a particular case (the sides are guided by the criterion of breaking through the weakest point when breaking through and holding defense points), the problem of distributing forces and facilities between tactical tasks (echelons) was solved according to two criteria: a) maximizing the probability of breaking through the defense point and the probability of defeating the enemy in depth defense, b) maximizing the minimum value of the named probabilities (the criterion of the guaranteed result). Awareness is an important aspect of combat operations. Several examples of reflexive games (games characterized by complex mutual awareness) and information management are considered. It is shown under what conditions information control increases the player’s payoff, and the optimal information control is found.