The inverse coefficient 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 r(0)u _x (0, t) = f(t), u(ℓ, t) = 0, from the Dirichlet boundary measured output ν(t) ≔ u(0, t), t ∈ (0, T] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q(x) in the interval [0, h(T/2)] and this solution belongs to C(0, h(T/2)) with T < T*, where h(z) is the root of the equation , . Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator , Φ ^f [q](t) ≔ u(0, t; q) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional as well as its Frchet differentiability. An explicit formula for the Frchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm.

回收的电位的逆系数问题q（X阻尼波方程），（X，吨）∈Ω _Ť ≔（0，ℓ）（0，Ť）除边界条件[R（0）Ü _X（0 ，吨）= ˚F（吨），ù（ℓ，吨）= 0，从Dirichlet边界测量的输出ν（吨）≔ ù（0，吨），吨∈（0，Ť ]被研究。提供了由特征以及沿着这些特征定义的子域中直接问题解决方案的规律性的详细微观局部分析。基于此分析，得出必要的规律性结果和能量估计。证明Dirichlet边界测得的输出唯一确定区间[0，h（T / 2）]中的电势q（x），并且该解属于C（0，h（T / 2）），其中T < T *，其中h（z）是方程的根，。此外，证明了全局唯一性定理。紧凑，诺伊曼到狄利克雷操作者的可逆性和利普希茨连续，Φ ^˚F [ q ]（吨）≔ ù（0，吨; q）证明。这使我们能够证明反问题的拟解的存在性，反问题定义为Tikhonov函数及其Frchet可微性的最小值。通过使用对应的伴随问题的唯一解，得出Frchet梯度的显式公式。所提出的方法导致非常有效的基于梯度的计算识别算法。