<p>This paper addresses an inverse problem concerning the reconstruction of the radiation coefficient in a Crocco-type degenerate parabolic equation using terminal observation data. The Crocco equation, derived from the Prandtl boundary layer equation, exhibits degenerate characteristics that complicate the analysis of the forward problem. First, by employing the elliptic regularization method, we establish the existence and uniqueness of weak solutions to the forward problem within a Sobolev space framework. Within an optimal control framework, the inverse problem is reformulated as an optimization problem. By adjusting the regularity of the penalty term, we investigate the influence of different penalty functions on the solution of the optimal control problem. The existence of a minimizer for the cost functional is proved, along with the necessary optimality conditions. Furthermore, the uniqueness and stability of the minimizer are derived from these necessary conditions. Finally, the forward problem is solved numerically using the alternating direction implicit method, and a gradient descent-based algorithm is designed to solve the inverse problem. Several representative numerical examples are provided, demonstrating that the reconstruction error is small and that the algorithm exhibits good robustness and accuracy.</p>