Uniqueness criterion in an inverse problem for an abstract differential equation with nonstationary inhomogeneous term

In a Banach space $E$, we consider the inverse problem $du(t)/dt=Au(t)+\phi(t)p$, $u(0)=u_0$, $u(T)=u_1$, with an unknown function $u(t)$ and an element $p\in E$. The operator $A$ is assumed linear and closed. In this paper, we establish minimal constraints on the function $\phi\in C([0,T])$ for which the uniqueness of the solution of the inverse problem is completely described in terms of the eigenvalues of the operator $A$.