## 4 Conclusions  
In this paper, a numerical method for solving the direct and inverse problems of the gas lift process in oil production, based on the method of conjugate equations, is developed and implemented. The key novelty of the method lies in the application of a variational formulation and the construction of a conjugate problem for calculating the gradient of the functional, which allows us to effectively solve the inverse problem of restoring the initial distribution of gas velocity for a given final state.  
Unlike traditional empirical or heuristic methods for optimizing gas lift processes, the proposed method is based on a rigorous mathematical framework and ensures controlled convergence. This allows us not only to increase the accuracy of calculations but also to reduce the volume of injected gas while maintaining the level of oil production, which is of great practical importance.  
The conducted numerical experiments confirmed the effectiveness of the method: high accuracy in approximating the target function was achieved in a limited number of iterations, while stable and reproducible results were obtained. The method can be used to optimize the operating parameters of gas lift wells and can also be adapted to more complex models, including multidimensional and nonlinear settings.  
The practical significance of the proposed method lies in its potential integration into software packages for managing oil production at gas lift wells. By restoring the optimal initial distribution of gas velocity, a more precise adjustment of the injection mode is ensured, which allows for the following benefits:  
* Reduce the volume of gas consumed while maintaining the production level.
* Adapt control to changing reservoir and well characteristics.
* Increase operating efficiency without the need to install additional sensors.
* Predict and minimize the risks of underloading or the over-consumption of gas.  
Thus, this study represents a significant contribution to the development of optimal control methods for problems described by hyperbolic equations and opens up opportuni  
Figure 5: Graph of the function \(Q_{n}(t,x)\) for different values of \(t\) with approximate initial conditions.  
ties for further applied developments in modeling oil and gas processes. The method can also serve as a decision support tool in intelligent oil field management systems, especially when implementing automation and digitalization in oil production.  
Conceptualization, S.E.K. and N.M.T.; methodology, A.K.T. and N.M.T.; software, A.K.T. and S.E.K.; validation, S.E.K. and A.K.T.; formal analysis, S.E.K. and A.K.T.; investigation, A.K.T.; resources, S.E.K. and A.K.T.; data curation, S.E.K. and A.K.T.; writing--original draft preparation, A.K.T.; writing--review and editing, A.K.T.; visualization, A.K.T.; supervision, N.M.T.; project administration, A.K.T. and N.M.T.; funding acquisition, A.K.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP22683374--\"Numerical solution of the multiphase dynamic model of the gas-lift process\").  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.  
The authors declare no conflicts of interest.