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),w(x)) under constraints du+ <sub>U</sub>(x)|<sub>x</sub> = f (x, t), (x, t) e Q = (0, i )x(0,T), (1) dt dx dx u(x,0) = u<sub>0</sub> (x),<sup>du (x,0)</sup> = u<sub>1</sub> (x),0 - x - £, (2) dt f^U = 0,f^U, = 0,0 - t - T, (3) dx dx u(x,T) = g(x),0 - x -I, here, f e L<sub>2</sub> (Q), u<sub>0</sub> e W\ [0,i], u<sub>1</sub> e L<sub>2</sub> (0,1), g e W,<sup>1</sup> [0,i] - are given functions. This problem is reduced to the following optimal control problem: find a function belonging to the set V = j^x) e W1 [0,t]: |о(x)| -M<sub>1</sub>,|о'^)| -M<sub>2</sub> a.e.on [0,(4) and minimizing the functional 1 <sup>1</sup> J (о) = - J[u (x,T;о) - g(x)]x (5) under constraints (1)-(3), where u (x, t; о) is a solution of problem (1)-(3) at a given w(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 J^My, (x,t) ((x) - о (x))dxdt > 0 Q <sup>dx</sup> where y, (x,t) is solution of the adjoint problem corresponding to the control о, = о, (x) d <sup>2</sup>y d <sup>2</sup>y d., _, _ -ZT<sup>-</sup>TT"(оу) = <sup>0,</sup>(x<sup>,</sup>t) e Q<sup>,</sup> dt dx dx v| t=<sub>T</sub> = 0,·<sup>дУ</sup>| <sub>t</sub>=<sub>T</sub> = u (x,T;о) - g(x),0 - x - i, dt ^=0 = 0,^U = 0,0 -1 - T dx dx is a necessary condition for optimality of the control о, = о, (x) e V of the problem (1)-(3), (4), (5) if it is fulfilled for all v e V.