  This paper introduces a Lyapunov-based control approach with homodyne
measurement. We study two filtering approaches: (i) the traditional quantum
filtering and (ii) a modified version of the extended Kalman filtering. We
examine both methods in order to directly estimate the evolution of the
coherence vector elements, using sequential homodyne current measurements. The
latter case explicitly addresses the dynamics of a stochastic master equation
with correlated noise, transformed into a state-space representation, ensuring
by construction the quantum properties of the estimated state variable. In
addition, we consider the case where the quantum-mechanical Hamiltonian is
unknown, and the system experiences uncertainties. In this case, we show as
expected that both filters lose performance, exhibiting large expected
estimation errors. To address this problem, we propose a simple multiple model
estimation scheme that can be directly applied to any of the studied filters.
We then reconstruct the estimated density operator \( \hat{\rho} \), describing
the full state of the system, and subject it to a control scheme. The proposed
switching-based Lyapunov control scheme, which is fed with \( \hat{\rho} \),
guarantees noise-to-state practically stable in probability of the desired
stationary target set with respect to the estimation error variance. We
demonstrate our approach's efficacy in stabilizing a qubit coupled to a leaky
cavity under homodyne detection in the presence of uncertainty in resonance
frequency.
