Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations (Q1326009)

If \(E\) is a real Banach space, the authors consider the abstract Cauchy problem \[ \begin{cases} u'(t)- Au(t)= \Phi(t) f\qquad (0\leq t\leq T),\\ u(0)= 0\end{cases}\tag{1} \] where \(T> 0\), \(A\) is a closed linear operator with domain \(D(A)\) dense in \(E\), \(f\in E\), and \(\Phi\in C^ 1([0, T]; B(E))\). If a certain operator \(\ell: C([0, T]; E)\mapsto E\) has a specific form, they prove that then, under some restrictions on \(A\) and \(\Phi\), the inverse problems of finding \(u(t)\) and \(f\in E\) satisfying (1) and condition \(\ell(u(t))= \psi\) \((\psi\in D(A))\), is well posed.