  Control by dissipation, or environment engineering, constitutes an important
methodology within quantum coherent control which was proposed to improve the
robustness and scalability of quantum control systems. The system-environment
coupling, often considered to be detrimental to quantum coherence, also
provides the means to steer the system to desired states. This paper aims to
develop the theory for engineering of the dissipation, based on a ground-state
Lyapunov stability analysis of open quantum systems via a Heisenberg-picture
approach. Algebraic conditions concerning the ground-state stability and
scalability of quantum systems are obtained. In particular, Lyapunov stability
conditions expressed as operator inequalities allow a purely algebraic
treatment of the environment engineering problem, which facilitates the
integration of quantum components into a large-scale quantum system and draws
an explicit connection to the classical theory of vector Lyapunov functions and
decomposition-aggregation methods for control of complex systems. The
implications of the results in relation to dissipative quantum computing and
state engineering are also discussed in this paper.
