Uniqueness of solution of the inverse problem with overdetermination of the raised type for an abstract second-order differential equation

We consider a linear inverse problem for a second-order abstract differential equation in a Banach space. The inhomogeneous term of the equation does not depend on time and is unknown. At the initial moment of time, the standard Cauchy conditions are given. An additional condition is specified at the final moment of time. This is a value of the second derivative of the main evolutionary function. For the studied problem, a uniqueness criterion of a solution is established. It is expressed in spectral terms. A simple sufficient condition for the solution uniqueness is noted. An example of the inverse problem for Poisson's equation in a cylindrical domain is considered.