Linear differential-algebraic equations in spaces of integrable functions (Q1826005)

An \(L^ 2\)-theory for linear time-varying differential-algebraic equations of the form \[ (\tilde Tx)(t):=A(t)x'(t)+B(t)x(t)=f(t),\quad t\in (a,b) \] is presented. It is assumed that the matrix pencil (A(t),B(t)) is of index 1. Sufficient conditions for the closability of \(\tilde T\) and the representation of the closed operator \(\tilde T\) are given. The main result states a sufficient condition when \(\tilde T\) is normally solvable. The question of ill-posed problems is tackled.