On the inverse problem of finding the right-hand side of wave equation with nonlocal condition

Recently, inverse problems for the differential equations have been intensively studied. Such problems arise in the various fields of mathematics, geophysics, seismology, astronomy, ecology, etc. In this paper, we propose an approach to solving the inverse problem for the wave equation. The search for the unknown right-hand side of the equation is reduced to the problem of minimizing the functional constructed using additional information. The gradient of the functional is calculated and the optimality condition is derived.
In the cylinder $\mathcal{Q}_T=\Omega\times(0,T)$, consider the problem
\begin{gather} \frac{\partial^2u}{\partial t^2}-\Delta u=\vartheta(x,t), \quad (x,t)\in\mathcal{Q}_T,\\ u(x,0)=\varphi_0(x), \frac{\partial u(x,0)}{\partial t}=\varphi_1(x), \quad x\in\Omega,\\ \frac{\partial u}{\partial \nu}\Big|_{S_T}=\int_\Omega K(x,y)u(y,t)dy,\quad (x,t)\in S_T, \end{gather}
where $\Omega\in R^n$ is a bounded domain with a smooth boundary $\partial\Omega$, $S_T=\partial\Omega\times(0,T)$ is the laterial surface of $\mathcal{Q}_T$, $\nu$ is an outward normal to $\partial\Omega$, $\varphi_0(x)\in W_2^1(\Omega)$, $\varphi_1(x)\in L_2(\Omega)$, $K(x,y)\in L_2(\Omega\times\Omega)$ are given functions, and $\vartheta(x,t)\in L_2(\mathcal{Q}_T)$ is the unknown function. To determine $\vartheta(x,t)$, we use the following additional information:
\begin{equation} u(x, T)=g(x), x\in\Omega, \text{ where }g(x)\in L_2(\Omega) \text{ is a given function.} \tag{4} \end{equation}
The problem is reduced to the following problem: minimize the functional
\begin{equation} J_0(\vartheta)=\frac12\int_\Omega(u(x,T;\vartheta)-g(x))^2dx\tag{5} \end{equation}
subject to (1)–(3), where $u(x,T;\vartheta)$ is a solution of problem (1)–(3) corresponding to $\vartheta(x,t)$ which is called a control. The solvability of problem (1)–(3), (5) is proved.
Consider the functional
\begin{equation} J_\alpha(\vartheta)=J_0(\vartheta)+\frac\alpha2\int_0^T\int_\Omega(\vartheta(x,t)-\omega(x,t))^2dx\,dt.\tag{6} \end{equation}
Then, the differential of this functional is calculated and the following theorem is proved:
Theorem. Under the considered conditions, for the optimality of the control $\vartheta_*=\vartheta_*(x,t)\in U_{ad}$ in the problem (1)–(3), (6) it is necessary that the inequality
\begin{equation} \int_0^T\int_\Omega(\alpha(\vartheta_*-\omega)-\psi(x,t;\vartheta_*))(\vartheta-\vartheta_*)dx\,dt\geqslant0\tag{7} \end{equation}
is fulfilled for all $\vartheta\in U_{ad}$.