Identifiability of operators for evolution equations in Banach spaces with an application to transport equations
In a Banach space \(X\) over the complex field \(C\), the equations \(du(t)/dt + A^ mu(t) = 0\) \((t \geq 0)\), \(u(0) = a^ m \in X\), \(dv(t)/dt + Av(t) = 0\) \((t \geq 0)\), \(v(0) = a \in X\), are considered. Here \(-A^ m\) and \(-A\) generate the \(C_ 0\)-semigroups \((\exp - tA^ m)_{t \geq 0}\) and \((\exp - tA)_{t \geq 0}\) respectively. The superscript \(m\) means quantities known as model systems. The problem of identifiability of operators and of initial values studied is the following: Let \(a^ m_ k\) \((1 \leq k \leq N)\) be known initial data and \(t_ 0 \geq 0\) be arbitrarily given. Under what conditions do the equalities \(\exp (- tA)a_ k = \exp (-tA^ m) a^ m_ k\) \((t \geq t_ 0\), \(1 \leq k \leq N)\) imply \(A = A^ m\) and \(a_ k = a^ m_ k\) \((1 \leq k \leq N)\)? Several abstract identifiability results are established by means of spectral theory for linear operators. As an application of these results, the identifiability of macroscopic total cross sections of transport equations is given.
macroscopic total cross sections of transport equations
scientific article; zbMATH DE number 663713