## 2 The Direct and Inverse Problems  
For the description of the dynamic model, we followed [2; 3; 4; 5]. We considered a simplified MHD Hartman flow with incompressible and Newtonian (constant viscosity) characteristics in 1D space as well as a case in which the fluid flowed between two parallel solid plates and where the velocity was perpendicular to the magnetic vector. Moreover, the pressure in the channel was supposed to be constant and the unit vectors of velocity,the mathematical model describing this 1D MHD flow system, could be derived from the viscous incompressible MHD equations [2; 3; 4; 5]. It reads as follows:  
\[\frac{\partial u}{\partial t}-\
u\frac{\partial^{2}u}{\partial x^{2} }=\beta(t)\frac{\partial B}{\partial x}-f(t), \tag{1}\] \[\frac{\partial B}{\partial t}-\
u_{m}\frac{\partial^{2}B}{ \partial x^{2}}=\beta(t)\frac{\partial u}{\partial x}. \tag{2}\]  
Here, the spatial variable \(x\) and time variable \(t\) belong to the set \((x,t)\in Q_{T}=\Omega\times(O,T)\), where \(\Omega=(0,1)\). In this context, \(u(x,t)\) denotes the flow velocity, \(B(x,t)\) is the magnetic field, \(f(t)\) is the pressure difference per unit length of the channel, and the function \(\beta(t)\) is a given function of the induction of the extended magnetic fields, which can be considered as the control input for the MHD flow. Additionally, \(\
u\) and \(\
u_{m}\) are two positive kinetic viscosity coefficients.  
In [3], it is assumed that the boundary conditions on the magnetic on the solid surfaces are zero velocity for the viscous fluid and zero boundary conditions for the magnetic field, corresponding to the continuity of the magnetic field strength. Thus, the boundary conditions for system (1) and (2) are given as follows:  
\[u(0,t)=u_{l}(t),\;\;u(1,t)=u_{r}(t), \tag{3}\] \[B(0,t)=B_{l}(t),\;\;B(1,t)=B_{r}(t). \tag{4}\]  
Furthermore, the initial conditions for the dynamic system (1), (2) are provided by  
\[u(x,0)=u_{0}(x),\;\;B(x,0)=B_{0}(x). \tag{5}\]  
In the form (1)-(5), we pose the _direct (forward) problem_, in which the right-hand sides, coefficients, boundary conditions, and initial conditions are given.  
Consider the inverse problem in which the coefficient \(\beta(t)\) and the source \(f(t)\) are unknowns, along with the over-specified total energy condition.  
\[\int_{0}^{1}u(x,t)dx=E(t), \tag{6}\]  
and total magnetic energy  
\[\int_{0}^{1}B(x,t)dx=F(t). \tag{7}\]  
Here, \(E(t)\) and \(F(t)\) are the given functions.  
Least squares is often used to define the closeness (objective functional). The resulting methods are referred to as the least squares output error criteria procedures; see, e.g., [3]. Although these procedures have many advantages, they also have drawbacks. For example, they are computationally intensive because iterative procedures (such as gradient-like search) are used. In addition, each step in the iteration reduces to solving a forward problem which involves the numerical solution of partial differential equations. Thus, there is a clear need to minimize the number of steps in the iteration, as well as to reduce the range of the global search. One way this can be achieved is to obtain an independent initial estimate of the structure of the control parameters. In the present work, our goal is to construct an efficient numerical procedure. We transform the identification problem for recovering time-dependent functions \(\beta(t),f(t)\) into a non-classical forward problem.