An inverse problem for the KdV equation with Neumann boundary measured data

In this paper, we consider an inverse coefficient problem for the linearized Korteweg-de Vries (KdV) equation u(t) + u(xxx) + (c(x)u)(x) = 0, with homogeneous boundary conditions u(0, t) = u(1, t) = u(x)(1, t) = 0, when the Neumann data g(t) := u(x)(0, t), t is an element of (0, T), is given as an available measured output at the boundary x = 0. The inverse problem is formulated as a minimum problem for the regularized Tikhonov functional J(alpha)(c) = 1.2 parallel to u(x)(0, center dot; c) - g parallel to(2)(L2(0, T)) + alpha/2 parallel to c' parallel to(2)(L2(0,1)) with Sobolev norm. Based on a priori estimates for theweak and regular weak solutions of the direct and adjoint problems, it is proved that the input-output operator is compact, which shows the ill-posedness of the inverse problem. Then Frechet differentiability of the Tikhonov functional and Lipschitz continuity of the Frechet gradient are proved. It is shown that the last result allows us to use an important advantage of gradient methods when the functional is from the class C-1,C-1(M). In the final part, an existence of a solution of the minimum problem for the regularized Tikhonov functional J(alpha)(c) is proved.