  This dissertation presents and prove the viability of a non-standard method
for controlling the state of a quantum system by modifying its boundary
conditions instead of relying on the action of external fields. The standard
approach to quantum control bases on the use of an external field to manipulate
the system. Some technological difficulties appear when controlling a quantum
system in this way, due to the complications of manipulating a system made of
few particles while maintaining the quantum correlations. As a consequence the
systems need to be kept at very low temperatures and the interactions have to
be performed very fast. The Quantum Control at the Boundary approach is
radically different to the standard one. Instead of seeking the control of the
quantum system by directly interacting with it through an external field, the
control is achieved by manipulating the boundary conditions of the system. The
spectrum of a quantum system, for instance an electron moving in a box, depends
on the boundary conditions imposed on it. Hence, a modification of such
boundary conditions modifies the state of the system allowing for its
manipulation and, eventually, its control. This kind of interaction is weaker,
which makes one to expect that it may help maintaining the quantum
correlations. For showing the viability of the Quantum Control at the Boundary
method, a family of boundary control systems on Quantum Circuits (a
generalization of quantum grahs) is introduced. Before being able to address
the problem of controllability, the problem of existence of solutions for the
Schr\"odinger equation with time-dependent boundary conditions is addressed.
The approximate controllability of the systems under study is proven using a
controllability result by T. Chambrion et al. (2009) and a stability result
which constitutes another original contribution of this dissertation.
