The inverse problem of recovering the potential q(x) in the damped wave equation , (x, t) ∈ Ω _T ≔ (0, ℓ) (0, T) subject to the boundary conditions u(0, t) = ν(t), u(ℓ, t) = 0, from the Neumann boundary measured output f(t) ≔ r(0)u _x (0, t), t ∈ (0, T] is studied. The approach proposed in this paper allows us to derive behavior of the direct problem solution in the subdomains defined by characteristics of the wave equation and along the characteristic lines, as well. Based on these results, a local existence theorem and the stability estimate are proved. The compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator are derived. Frchet differentiability of the Tikhonov functional is proved and an explicit gradient formula is derived by means of an appropriate adjoint problem. It is proved that this gradient is Lipschitz continuous.

回收的电位的逆问题q（X阻尼波方程），（X，吨）∈Ω _Ť ≔（0，ℓ）（0，Ť）除边界条件ù（0，吨）= ν（吨），ù（ℓ，吨）= 0，从Neumann边界测量的输出˚F（吨）≔ - [R（0）ü _X（0，吨），吨∈（0，Ť ]被研究。本文提出的方法使我们能够在由波动方程的特征以及沿着特征线定义的子域中导出直接问题解的行为。基于这些结果，证明了局部存在性定理和稳定性估计。推导了Dirichlet-to-Neumann算子的紧致性和Lipschitz连续性。证明了Tikhonov函数的Frchet可微性，并通过适当的伴随问题导出了一个明确的梯度公式。证明该梯度是Lipschitz连续的。