1998, VOLUME 4, NUMBER 2, PAGES 691-708

On the solvability of linear inverse problem with final overdetermination in a Banach space of L1-type

I. V. Tikhonov


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Given T > 0 we consider the inverse problem in a Banach space E

du(t)/dt = Au(t) + Φ(t) f,  0 ≤ t ≤ T,
u(0) = u0,  u(T) = u1,   u0, u1 ∈ D(A)

where the element f ∈ E is unknown. Our main result may be written as follows: Let E=L1(X, μ ) and let A be the infinitesimal generator of a C0 semigroup U(t) on L1 (X, μ) satisfying ||U(t)|| < 1 for t > 0. Let Φ (t) be defined by

(Φ(t)) (x) = φ (x,t) f(x)

where φ ∈ C1([0,T];L(X, μ )). Suppose that φ (x,t) ≥ 0, ∂ φ (x,t)/ ∂ t ≥ 0 and μ-inf φ (x,T) > 0. Then for each pair u0,u1 ∈ D(A) the inverse problem has a unique solution f ∈ L1(X, μ ), i. e., there exists a unique f ∈ L1(X, μ ) such that the corresponding function

u(t)=U(t)u0+ ∫ 0tU(t-s) Φ (s)f ds,    0 ≤ t ≤ T,

satisfies the final condition u(T)=u1. Moreover, ||f|| ≤ C(||Au0||+||Au1||) with the constant C > 0 computing in the explicit form. To illustrate the results we present three examples: the linear inhomogeneous system of ODE, the heat equation in $ \mathbb{R}^n $, and the one-dimensional "transport equation".

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Last modified: June 17, 1998