## 1 Introduction
In [10], a method for optimal coverage of the territory by sensors with different coverage radii was proposed. To solve this problem, the authors modified the least squares method. The criterion for optimality is the maximum coverage of the territory with a minimum intersection of coverage areas. The existence of the solution of the described optimization problem is not proven. In the manuscript [11], models and information technologies of coverage of the territory by sensors with energy consumption optimization are described. A modification of the method for constructing energy-efficient sensor networks using static and dynamic sensors is presented in article [12]. A Method for Maximum Coverage of the Territory by Sensors with Minimization of Cost is described in paper [13]. In the described works [11; 12; 13], authors find optimal sensor network parameters by a solution of multicritertial optimization problems, but the existence of the solution of these optimization problems is not proven. Attention should also be paid to the use of distributed methods to solve current problems that arise in the design and use of wireless sensor networks. Thus, in work [14], an energy-aware method for selecting cluster heads in wireless sensor networks is presented. To select an appropriate cluster head, authors first model this problem by using multi-factor decision-making according to the four factors, including energy, mobility, distance to center and the length of data queues. Then, authors use the Cluster Splitting Process algorithm and the Analytical Hierarchy Process method in order to provide a new method to solve this problem. However, the authors do not take into account the radius of coverage of each sensor as a factor affecting energy consumption and the intersection of sensor coverage areas for data exchange between sensors. In the article [15], a wireless power transfer networks' meanwhile maximizing the transmitted power requirement as an optimization problem is presented. In order to provide an approximated solution to this problem, the authors introduce a dual ascent-like distributed charging algorithm with its in-depth theoretical analysis, but the existence of the solution of the formulated optimization problems is not proved. In reference [16], a multi-agent distributed solution for linear programming problems with time-invariant box constraints on the decision variables and possibly time-varying inequality constraints is proposed. Such a class of linear programming problems is relevant in different multi-agent smart systems. In the proposed approach, each agent computes only a single or a few of decision variables, while convergence to the optimal solution for the overall problem is guaranteed. Table 1 presents a comparison of the studies found in the literature review, the main focus/outcome of the study, and the factors that were not taken into account. Based on the reviewed studies, it can be concluded that the problem of energy efficiency of sensor networks is very relevant. Different methods and approaches are used to solve this problem; in particular, they formulate this problem in the form of an optimization problem. In the theory of optimization, a necessary step in solving problems is proving the existence of a solution to the formulated problem. Unfortunately, in most modern studies, the authors do not prove the existence of a solution to the described problems. That is why, based on the studies reviewed above, it can be concluded that proving the existence of solutions of optimization problems for finding optimal parameters of sensor networks is an urgent problem.