## 2 Materials and Methods  
Let's consider the problem of finding the radii of coverage of the sensors of the sensor network, which achieves the minimization of energy consumption under the condition of the intersection of coverage areas. According to [11], this problem has the form (1)-(5):  
\[E(r_{1},r_{2},\ldots,r_{n})\rightarrow\min_{r\in G}\, \tag{1}\]  
\[G: \begin{cases}r_{1}\in\left[r_{1}^{min};r_{2}^{max}\right]\\ r_{2}\in\left[r_{2}^{min};r_{2}^{max}\right],\\ r_{n}\in\left[r_{n}^{min};r_{n}^{max}\right]\end{cases} , \tag{2}\]  
\[a_{ij}\bigg{(}r_{i}+r_{j}-\sqrt{\big{(}x_{i}-x_{j}\big{)}^{2}+\big{(}y_{i}-y_{ j}\big{)}^{2}}-c\bigg{)}\leq 0, \tag{3}\]  
\[-a_{ij}<0, \tag{4}\]  
\[i=\overline{1,n},\ j=\overline{i+1,n}, \tag{5}\]  
where (1)--the objective function, (2)--the area of definition of the function, (3)--the conditions for ensuring the minimum allowable cross section of a pair of sensors whose coverage areas intersect, (4)--intersection matrix elements \(a_{ij}=\left\{ \begin{array}{l}1,\ l_{ij}<r_{i}+r_{j}\\ 0,l_{ij}\geq r_{i}+r_{j}\end{array} ,\ l_{ij}\)--Euclidean distance between sensors \(i\) and \(j\), and \(n\)--count of sensors. The energy consumption of the network (1) calculates as the sum of the energy consumption of network elements:  
\[E(r_{1},r_{2},\ldots,r_{n})=\sum_{i=1}^{n}(Er(r_{i})+Es_{i}+Ea_{i}). \tag{6}\]  
According to [12], the energy consumption of each sensor is calculated as:  
\[E=Er+Es+Ea, \tag{7}\]  
where \(Er\)--energy consumption of data reception, \(Es\)--energy consumption of data transmission, and \(Ea\)--energy consumption monitoring of the covered area. In turn, according to [17], energy consumption monitoring will be calculated as:  
\[Ea=r^{2}. \tag{8}\]  
For the existence of the solution of the described problem, the goal function must be convex, and the domain of the function must be compact. Let us prove the existence of a solution to the formulated problem. **Lemma 1**. : _The function \(E(r_{1},r_{2},\ldots,r_{n})\) reaches its minimum value in the domain \(G\)._  
**Proof of Lemma 1. ** The function \(E(r_{1},r_{2},\ldots,r_{n})\) is continuous as the sum of continuous functions (6). The domain \(G\) is bounded because all variables have minimum and maximum values of \(r_{i}\in\left[r_{i}^{min};r_{i}^{max}\right]\), \(i=\overline{1,n}\). Additionally, the domain \(G\) is closed, since for all variables the segment \(\left[r_{i}^{min};r_{i}^{max}\right]\) is closed. Thus, the domain \(G\) is compact, and, according to [PERSON]'s theorem [18], the function \(E(r_{1},r_{2},\ldots,r_{n})\) reaches its minimum in the domain \(G\). Lemma is proved. \(\square\)  
**Theorem 1**. : _The problems (1)-(5) has a solution._  
**Proof of Theorem 1. ** The function \(E(r_{1},r_{2},\ldots,r_{n})\) is convex, continuous and differentiated. According to Lemma 1, the domain \(G\) is compact; then, according to the Kuhn-Tuckertheorem [19], there is a saddle point of the Lagrange function, and it is an extremum. Thus, problems (1)-(5) has a solution. The theorem is proved. \(\square\)  
Since the function \(E(r_{1},r_{2},\ldots,r_{n})\) is convex and differentiated, we use the gradient descent method to find the minimum of the objective function. The solution of problems (1)-(5) is the vector \(r=(r_{1},r_{2},\ldots,r_{n})\), the elements of which are the radii of coverage of the corresponding sensors, which minimizes energy consumption at a given level of the intersection of coverage areas. According to [13], the problem of minimizing the cost of the sensor network while maximizing the coverage is formulated in the form (9)-(11):  
\[\left\{ \begin{array}{l}P(r,c)\to min\\ S_{uncovered}(r,c)\to min\end{array} \right., \tag{9}\]  
\[r_{min}\leq r\leq r_{max}, \tag{10}\]  
\[0\leq c\leq c_{max}, \tag{11}\]  
where \(r\)--coverage radius, \(c\)--intersection level, \(S_{uncovered}=(2r-c)^{2}-\pi r^{2},r_{min}\)--minimum coverage radius and \(r_{max}\)--maximum coverage radius.