Determination of a parameter of a parabolic equation in a Hilbert structure (Q1893650)

The author considers the inverse problem of determining a pair of functions \(u\in C^1([0, T]; X)\cap C([0, T]; {\mathcal D}(A))\) and \(p\in X\) from \[ u'(t)= Au(t)+ \Phi(t)p+ F(t),\;0\leq t\leq T,\;u(0)= u_0,\;u(T)= u_1, \] where the linear closed operator \(A\) with the dense domain \({\mathcal D}(A)\) is given in a Banach space \(X\). In the case that \(X\) is a Hilbert structure, conditions on the operator \(A\) and the function \(\phi\) are formulated under which there exists a unique solution of the inverse problem for any admissible data \(u_0\), \(u_1\), \(F\). This statement is proved using the theory of semigroups. An application of the result is given.