This dissertation analyzes a nonlinear hyperbolic PDE-constrained optimization problem. Motivated by applications in Full Waveform Inversion, our central goal is to reconstruct the wave speed parameter entering the acoustic wave equation in the coefficient of the second-order time derivative of the acoustic pressure. Starting with the first- and second-order analysis, we prove the well-definedness of the problem and establish corresponding necessary and sufficient optimality conditions. These findings lay the foundation for investigating the application of the Sequential Quadratic Programming method. Here, a broad extension of the parabolic techniques is required due to the hyperbolicity and the bilinear character of the underlying partial differential equation. Based on a two-step estimation process, we show the well-posedness and R-superlinear convergence of the algorithm. Furthermore, the present thesis includes the numerical analysis of a fully discrete approximation of the optimization problem, consisting of a Finite Element discretization in space and a leapfrog time-stepping. Building upon a stability analysis, we prove a convergence result regarding first-order necessary optimality conditions. Moreover, we demonstrate that for every local minimizer of the original problem that satisfies a reasonable growth condition, there is a sequence of locally optimal solutions to the discrete problems that converges to this minimizer. The document concludes with numerical experiments based on synthetic configurations with nonsmooth data, which illustrate the performance and effectiveness of the presented approach.