Identifications for General Degenerate Problems of Hyperbolic Type in Hilbert Spaces

In a Hilbert space X, we consider the abstract problemM∗ddt(My(t))=Ly(t)+f(t)z,0≤t≤τ,My(0)=My0 where L is a closed linear operator in X and M ∈ ℒ(X) is not necessarily invertible, z ∈ X. Given the additional information Φ[My(t)] = g(t) with Φ ∈ X^*, g ∈ C¹([0, τ];ℂ), we are concerned with the determination of the conditions under which we can identify f ∈ C([0, τ];ℂ) such that y be a strict solution to the abstract problem, i.e., My ∈ C¹([0, τ];X), Ly ∈ C([0, τ];X). A similar problem is considered for general second-order equations in time. Various examples of these general problems are given.