I came across an equation of the following type: Let $V$ and $H$ be two Hilbert spaces such that $V\hookrightarrow H$ compactly and densely. Introduce the Gelfand triple $V\hookrightarrow H \hookrightarrow V^{\prime}$. Consider \begin{equation} P^{\prime}(t) = A^{\ast}(t) P(t) + P(t) A(t) + S(t), \end{equation} where $A: V\to V^{\prime}$, $S:H\to H$ bounded, and $A^{\ast}$ denotes the adjoint of $A$ in $H$ (as an operator from $V$ to $V^{\prime}$).
This equation is similar to the Riccati equation that arises in linear quadratic optimal control, but it is easier, since it doesn't contain the quadratic term. I have found some literature on the solution of the Riccati equation, mostly considering the mild approach. Do you know of any literature solving these kind of equations using the variational approach?
Any comment is appreciated!
Best, Luke