(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Abstract

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

u(0) = u0,  u(T) = u1,   u0, u1 ∈ D(A)

where the element $f \in E$ is unknown. Our main result may be written as follows: Let $E=L1\left(X, \mu \right)$ and let $A$ be the infinitesimal generator of a $C$0 semigroup $U\left(t\right)$ on $L1\left(X, \mu \right)$ satisfying $||U\left(t\right)|| < 1$ for $t > 0$. Let $\Phi \left(t\right)$ be defined by

$\left(\Phi \left(t\right)\right) \left(x\right) = \phi \left(x,t\right) f\left(x\right)$

where $\phi \in C1\left(\left[0,T\right];L\infty \left(X, \mu \right)\right)$. Suppose that $\phi \left(x,t\right) \ge 0$, $\partial \phi \left(x,t\right)/ \partial t \ge 0$ and $\mu$-$inf \phi \left(x,T\right) > 0$. Then for each pair $u$0,u1 ∈ D(A) the inverse problem has a unique solution $f \in L1\left(X, \mu \right)$, i. e., there exists a unique $f \in L1\left(X, \mu \right)$ such that the corresponding function

$u\left(t\right)=U\left(t\right)u$0+ ∫ 0tU(t-s) Φ (s)f ds,    0 ≤ t ≤ T,

satisfies the final condition $u\left(T\right)=u$1. Moreover, $||f|| \le C\left(||Au$0||+||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 $\mathbb\left\{R\right\}^n$, and the one-dimensional "transport equation".

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