This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the hyperbolic character and the second-order bilinear structure in the governing wave equation. While the control parameter is discretized using the piecewise constant elements, the state discretization is realized through an auxiliary first-order system along with the leapfrog time-stepping method and continuous piecewise linear elements. The resulting fully discrete minimization problem ($\text{P}_h$) is shown to be well-defined. Furthermore, building upon a suitable CFL-condition, we prove stability and uniform convergence of the state discretization. Our final result is the strong convergence result for ($\text{P}_h$) in the following sense: Given a local minimizer $\overline \nu$ of (P) satisfying a reasonable growth condition, there exists a sequence of local minimizers of ($\text{P}_h$) converging strongly towards $\overline \nu$.

翻译：本文探讨了应用于声学全波形反演的非线性双曲偏微分方程约束优化问题（P）的完全离散逼近。该优化问题的主要复杂性源于控制波动方程的双曲特性与二阶双线性结构。控制参数采用分段常数元进行离散化，而状态离散化则通过辅助一阶系统结合蛙跳时间步进方法与连续分段线性元实现。所得完全离散最小化问题（$\text{P}_h$）被证明是良定的。此外，基于合适的CFL条件，我们证明了状态离散化的稳定性与一致收敛性。最终结论为（$\text{P}_h$）在以下意义下的强收敛结果：给定满足合理增长条件的（P）的局部极小化子$\overline \nu$，存在一列（$\text{P}_h$）的局部极小化子强收敛于$\overline \nu$。