Reduction of the acoustic inverse problem to an optimal control problem and its investigation

In this paper, the coefficient inverse problem for the one-dimensional acoustic equation is considered. The problem is reduced to an optimal control problem. In the new problem, the existence theorems are proved, necessary conditions of optimality are derived, differentiability of the functional is shown, and an iteration algorithm for finding the solution of the optimal control problem based on the gradient projection method is proposed.
We consider the problem of determining a pair of functions $(u (x,t),\upsilon(x))$ under constraints
\begin{gather} \frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}+\upsilon(x)\frac{\partial u}{\partial x}=f(x,t),\quad (x,t)\in\mathcal{Q}\equiv(0,\ell)\times(0,T),\\ u(x,0)=u_0(x), \frac{\partial u(x,0)}{\partial t}=u_1(x), \quad 0\leqslant x\leqslant\ell,\\ \frac{\partial u}{\partial x}\mid_{x=0}=0, \frac{\partial u}{\partial x}\mid_{x=\ell}=0, \quad 0\leqslant t\leqslant T,\\ u(x,T)=g(x), \quad 0\leqslant x\leqslant\ell,\notag \end{gather}
here, $f\in L_2(\mathcal{Q})$, $u_0\in W_2^1[0,\ell]$, $u_1\in L_2(0,\ell)$, $g\in W_2^1[0,\ell]$ — are given functions.
This problem is reduced to the following optimal control problem: find a function belonging to the set
\begin{equation} V=\left\{\upsilon(x)\in \overset{0}{W_2^1}[0,\ell]: |\upsilon(x)|\leqslant M_1, |\upsilon'(x)|\leqslant M_2 \text{ a.e.on }[0,\ell]\right\}, \end{equation}
and minimizing the functional
\begin{equation} J(\upsilon)=\frac12\int_0^\ell[u(x, T;\upsilon)-g(x)]^2dx \end{equation}
under constraints (1)–(3), where $u(x,t;\upsilon)$ is a solution of problem (1)–(3) at a given $\upsilon(x)$, which is called a control. The solvability of problem (1)–(3), (4), (5) is proved.
Then, the differential of the functional is calculated and the following theorem is proved.
Theorem. Under the conditions considered above, the inequality
$$ \int_{\mathcal{Q}}\frac{\partial u_*(x,t)}{\partial x}\psi_*(x,t)(\upsilon(x)-\upsilon_*(x))dxdt\geqslant 0 $$
where $\psi_*(x,t)$ is solution of the adjoint problem corresponding to the control $\upsilon_*=\upsilon_*(x)$:
\begin{gather*} \frac{\partial^2\psi}{\partial t^2}-\frac{\partial^2\psi}{\partial x^2}-\frac{\partial}{\partial x}(\upsilon\psi)=0, \quad (x,t)\in\mathcal{Q},\\ \psi\mid_{t=T}=0, \quad \frac{\partial\psi}{\partial t}\mid_{t=T}=u(x,T;\upsilon)-g(x), \quad 0\leqslant x\leqslant\ell,\\ \frac{\partial\psi}{\partial x}\mid_{x=0}=0, \quad\frac{\partial\psi}{\partial x}\mid_{x=\ell}=0, \quad 0\leqslant t\leqslant T \end{gather*}
is a necessary condition for optimality of the control $\upsilon_*=\upsilon_*(x)\in V$ of the problem (1)–(3), (4), (5) if it is fulfilled for all $v\in V$.