ABSTRACT
In this article we are concerned with an identification problem for first order linear systems extending the theory and methods discussed in [7] and [1]. See also [2] and [9]. Related nonsingular results were obtained in [11] under different additional conditions even in the regular case. There is a wide literature on inverse problems motivated by applied sciences. We refer to [11] for an extended list of references. Inverse problems for degenerate differential and integrodifferential equations are a new branch of research. Very recent results have been obtained in [7], [5] and [6] relative to identification problems for degenerate integrodifferential equations. Here we treat similar equations without the integral term and this allows us to lower the required regularity in time of the data by one. The singular case for infinitely differentiable semigroups and second order equations in time will be treated in some forthcoming papers. The contents of the paper are as follows. In Section 2 we present the non-
singular case, precisely, we consider the problem
u′(t) +Au(t) = f(t)z , 0 ≤ t ≤ τ , u(0) = u0 , Φ[u(t)] = g(t) , 0 ≤ t ≤ τ ,
where −A generates an analytic semigroup in X, X being a Banach space, Φ ∈ X∗, g ∈ C1([0, τ ],R), τ > 0 fixed, u0, z ∈ D(A) and the pair (u, f) ∈ C1+θ([0, τ ];X) × Cθ([0, τ ];R), θ ∈ (0, 1), is to be found. Here Cθ([0, τ ];X) denotes the space of all X-valued Ho¨lder-continuous functions on [0, τ ] with exponent θ, and
C1+θ([0, τ ];X) = {u ∈ C1([0, τ ];X); u′ ∈ Cθ([0, τ ];X)}.
