FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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 $E$

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

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

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

All articles are published in Russian.

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Last modified: June 17, 1998