  We investigate a time and energy minimization optimal control problem for
open quantum systems, whose dynamics is governed through the Lindblad (or
Gorini-Kossakowski-Sudarshan-Lindblad) master equation. The dissipation is
Markovian time-independent, and the control is governed by the Hamiltonian of a
quantum-mechanical system. We are specifically interested to study the purity
in a dissipative system constrained by state and control inputs. The idea for
solving this problem is by the combination of two following techniques. We deal
with the state constraints through Gamkrelidze revisited method, while handling
control constraints through the idea of saturation functions and system
extensions. This is the first time that quantum purity conservation is
formulated in such framework. We obtain the necessary conditions of optimality
through the Pontryagin Minimum Principle. Finally, the resulted boundary value
problem is solved by a Physics-Informed Neural Network (PINN) approach. The
exploited Pontryagin PINN technique is also new in quantum control context. We
show that these PINNs play an effective role in learning optimal control
actions.
