  We study the problem of optimal control of dissipative quantum dynamics.
Although under most circumstances dissipation leads to an increase in entropy
(or a decrease in purity) of the system, there is an important class of
problems for which dissipation with external control can decrease the entropy
(or increase the purity) of the system. An important example is laser cooling.
In such systems, there is an interplay of the Hamiltonian part of the dynamics,
which is controllable and the dissipative part of the dynamics, which is
uncontrollable. The strategy is to control the Hamiltonian portion of the
evolution in such a way that the dissipation causes the purity of the system to
increase rather than decrease. The goal of this paper is to find the strategy
that leads to maximal purity at the final time. Under the assumption that
Hamiltonian control is complete and arbitrarily fast, we provide a general
framework by which to calculate optimal cooling strategies. These assumptions
lead to a great simplification, in which the control problem can be
reformulated in terms of the spectrum of eigenvalues of $\rho$, rather than
$\rho$ itself. By combining this formulation with the Hamilton-Jacobi-Bellman
theorem we are able to obtain an equation for the globaly optimal cooling
strategy in terms of the spectrum of the density matrix. For the three-level
$\Lambda$ system, we provide a complete analytic solution for the optimal
cooling strategy. For this system it is found that the optimal strategy does
not exploit system coherences and is a 'greedy' strategy, in which the purity
is increased maximally at each instant.
