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

Abstract

AbstractIn 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\in(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 {\mathcal{J}_{\alpha}(c)=\frac{1}{2}\|u_{x}(0,\cdot\,;c)-g\|^{2}_{L^{2}(0,T)}+% \frac{\alpha}{2}\|c^{\prime}\|^{2}_{L^{2}(0,1)}} with Sobolev norm. Based on a priori estimates for the weak 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 Fréchet differentiability of the Tikhonov functional and Lipschitz continuity of the Fréchet 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^{1,1}(\mathcal{M})}. In the final part, an existence of a solution of the minimum problem for the regularized Tikhonov functional {\mathcal{J}_{\alpha}(c)} is proved.