  The core problem in optimal control theory applied to quantum systems is to
determine the temporal shape of an applied field in order to maximize the
expectation of value of some physical observable. The functional which maps the
control field into a given value of the observable defines a Quantum Control
Landscape (QCL). Studying the topological and structural features of these
landscapes is of critical importance for understanding the process of finding
the optimal fields required to effectively control the system, specially when
external constraints are placed on both the field $\epsilon(t)$ and the
available control duration $T$. In this work we analyze the rich structure of
the $QCL$ of the paradigmatic Landau-Zener two-level model, studying several
features of the optimized solutions, such as their abundance, spatial
distribution and fidelities. We also inspect the optimization trajectories in
parameter space. We are able rationalize several geometrical and topological
aspects of the QCL of this simple model and the effects produced by the
constraints. Our study opens the door for a deeper understanding of the QCL of
general quantum systems.
