Justification of the steepest descent method for the integral statement of an inverse problem for a hyperbolic equation (Q2760723)

Consider the inverse problem of finding a solution \(u(x,t)\) and a coefficient \(q(x)\) such that NEWLINE\[NEWLINE \begin{gathered} u_{tt}=u_{xx}-q(x)u, \quad x\in {\mathbb R},\;t>0, \\ u(x,0)=q(x), \quad u_t(x,0)=0,\quad x\in {\mathbb R}, \\ u(0,t)=f(t), \quad u_x(0,t)=0,\quad t\geq 0. \end{gathered} NEWLINE\]NEWLINE Using the d'Alembert formula, the authors transform this problem to the problem of solving a nonlinear integral equation. The solution to this equation, in turn, is defined as the function minimizing some generally unavailable objective functional. The authors study this optimization problem by the steepest descent method and estimate the rate of convergence in the mean.