FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

I. V. Tikhonov

Abstract

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

\begin{gather*}

du(t)/dt=Au(t)+\Phi(t)f,\quad 0\le t\le T,\\

u(0)=u_0,\ \ u(T)=u_1,\quad u_0,u_1 \in D(A)

\end{gather*}

where the element $f\in E$ is unknown.
Our main result may be written as follows:

Let $E=L^1(X,\mu)$ and let $A$ be the infinitesimal generator
of a $C_0$ semigroup $U(t)$ on $L^1(X,\mu)$ satisfying
$\|U(t)\|<1$ for $t>0$ .
Let $\Phi(t)$ be defined by

$$

\big(\Phi(t)f\big)(x)=\varphi(x,t)\cdot f(x)

$$

where $\varphi\in C^1([0,T];L^\infty(X,\mu))$ .
Suppose that
$\varphi(x,t)\ge 0$ , $\partial\varphi(x,t)/\partial t\ge 0$
and $\mu\hbox{-}\inf\varphi(x,T)>0$ .
Then for each pair $u_0, u_1\in D(A)$ the inverse problem
has a unique solution $f\in L^1(X,\mu)$ ,
i. e., there exists a unique $f\in L^1(X,\mu)$
such that the corresponding function

$$

u(t)=U(t)u_0 +\int\limits_0^t U(t-s)\Phi(s)f\,ds, \quad 0\le t\le T,

$$

satisfies the final condition $u(T)=u_1$ .
Moreover, $\|f\|\le C(\|Au_0\|+\|Au_1\|)$ 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