from simulators such as PIPESIM is found, for example, in [2], and the problems related to distributing gas injection between wells are solved using mathematical programming methods and evolutionary algorithms [3].  
Modern methods for solving inverse problems arising in mathematical physics involve reducing them to optimal control problems with partial differential equations as constraints. The classical theory of such problems was developed by [PERSON] [4; 5], and the method of conjugate equations, based on the Lagrange identity and the duality principle, was studied in detail in the works of [PERSON] [6], [PERSON] [7] and other authors. The conjugate problem method allows us to effectively calculate the gradient of the objective functional, which makes it possible to apply gradient optimization methods [8; 9].  
Additional interest in the application of the conjugate equation method is due to its successful use in various fields--from string vibration control [8] to numerical solutions of acoustic problems and problems involving regularization methods [10; 11; 12; 13; 14; 15; 16]. An example of this work is study [17], which deals with the modeling and optimization of an industrial rotary kiln (RHF) used for processing solid metallurgical waste. In the study, a numerical model is built based on heat transfer and gas dynamics equations, and then a surrogate model is built using machine learning methods. Optimization is performed using industrial data to minimize energy costs. This methodology, which combines direct modeling and numerical optimization methods integrated with real data, is of great interest and can be adapted for gas lift process problems, where it is also necessary to determine the optimal initial pressure distribution or gas flow rate.  
A number of papers have proposed effective numerical methods for solving applied problems in mathematical physics. In [18], an algorithm was developed for solving the inverse problem for the acoustics equation using the gradient method, which ensures high accuracy of reconstruction of unknown parameters. In [19], an approach to numerical modeling of the transport of pollutants in the atmosphere is presented, which makes it possible to increase the accuracy of the forecast in an industrial environment. Both works demonstrate the importance of using modern computational methods to solve inverse and predictive problems in various fields.  
Despite the simplicity of the mathematical model used, the novelty of this work lies in the formalization and numerical solution of the inverse problem of determining the initial gas velocity function, \(G^{0}(x)\in H^{1}(0,2l)\), for a given flow rate distribution, \(Q^{1}(x)\), at time \(t=T\). Such a formulation is rarely considered in relation to gas lift processes. In this work, a numerical algorithm is developed based on the method of conjugate equations, the duality principle, and the Landweber method, which allows for the effective solution of such problems while taking into account the design features of the well (two zones with different characteristics and transition conditions at the point \(x=l\)).  
The aim of this work is to reconstruct the initial distribution of gas velocity in a system described by hyperbolic equations using additional information about the state of the system at the final moment in time. This task arises when it is necessary to clarify the initial conditions for modeling the gas lift process, particularly when direct measurements at the depth of the well are difficult or impossible.  
Unlike existing methods, this paper uses the conjugate equation method, which allows us to effectively find the gradient of the target functional and implement an iterative optimization procedure. This approach ensures high accuracy in solving the inverse problem with minimal computational costs. The obtained results confirm the applicability of the developed numerical method for restoring hidden parameters and optimizing the operating conditions of gas lift wells.