Identifications for general degenerate problems of hyperbolic type in Hilbert spaces

In a Hilbert space $X$, we consider the abstract problem
\begin{align*} &M^*\frac d{dt}(My(t))=Ly(t)+f(t)z,\quad0\le t\le\tau,\\ &My(0)=My_0, \end{align*}
where $L$ is a closed linear operator in $X$ and $M\in\mathcal L(X)$ is not necessarily invertible, $z\in X$. Given the additional information $\Phi[My(t)]=g(t)$ wuth $\Phi\in X^*$, $g\in C^1([0,\tau];\mathbb C)$. We are concerned with the determination of the conditions under which we can identify $f\in C([0,\tau];\mathbb C)$ such that $y$ be a strict solution to the abstract problem, i.e., $My\in C^1([0,\tau];X)$, $Ly\in C([0,\tau];X)$. A similar problem is considered for general second order equations in time. Various examples of these general problems are given.