On the solvability of linear inverse problem with final overdetermination in a Banach space of $L^1$-type

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 (cf. theorem 2): 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 $\varphi(x,t)\ge0$, $\partial\varphi(x,t)/\partial t\ge0$ and $\mu$-$\inf\varphi(x,T)>0$. 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 (see formulas (9), (11)). An abstract version of this assertion is given in theorem 1. 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”.