Consider a continuous Kalman filter running backward in time as desired in a "two-filter" smoother. What would be the form of Lyapunov equation for this backward-time filter?
Given a system: dx/dt = Fx + Gv, and, say, a forward Kalman filter: dX/dt = CX + Dx + Ew (where X is the filtered esimate of the state x and v, w are white noises), the augmented system would be:
d/dt [x X]' = A [x X]' + B [v w]'
Then, the (steady-state) Lyapunov equation for the augmented (forward-time) system is: AP + PA' + BB' = 0.
In backward-time, the system modifies to: -dx/dt = Fx + Gv, and accordingly the (backward) Kalman filter may be obtained using standard Riccati equation approach. How would the augmented system be obtained to be used for Lyapunov equation? What is the form of the Lyapunov equation in this case, if different from the forward-case mentioned earlier? I need to be able to use Lyapunov equation to derive the state-covariance matrix and the error-covariance from that.