  We propose a Lyapunov control design to achieve specific (or a family of)
unitary time-evolution operators, i.e., quantum gates in the Schr\"{o}dinger
picture by tracking control. Two examples are presented. In the first, we
illustrate how to realize the Hadamard gate in a single-qubit system, while in
the second, the controlled-NOT (CNOT) gate is implemented in two-qubit systems
with the Ising and Heisenberg interactions. Furthermore, we demonstrate that
the control can drive the time-evolution operator into the local equivalence
class of the CNOT gate and the operator keeps in this class forever with the
existence of Ising coupling.
