Consider the multi-dimensional linear filtering problem with no noise in the system:
$$\text{(system)} \qquad dX_{t}=F(t)X_{t}dt;$$
$$ X_{t}\in \mathbf{R}^{n} , F(t)\in \mathbf{R}^{n\times n}$$
$$\text{(observations)} \qquad dZ_{t}=G(t)X_{t}dt+D(t)dV_{t};$$
$$G(t)\in \mathbf{R}^{m\times n}, D(t)\in \mathbf{R}^{m\times r}$$
Then if we are to assume that $S(t)$ is nonsingular and define $R(t) = S(t)−1$ , how can we prove that $R(t)$ satisfies the Lyapunov equation
$$R'(t)=-R(t)F(t)-F(t)^{T}R(t)+G(t)^{T}(D(t)D(t)^{T})^{-1}G(t)$$