Abstract

Linear DAEs of the form \[ A(t)(D(t)x(t))’+B(t)x(t)=0 \] and their inhomogeneous counterparts were introduced in the seminal paper of \textit{K. Balla} and \textit{R. März} [Z. Anal. Anwend. 21, 783-802 (2002; Zbl 1024.34002)]. Such a DAE is said to be properly stated if \(A\) and \(D\) are well-matched in a certain sense; in this case, the adjoint problem \[ -D^*(t)(A^*(t)x(t))’+B^*(t)x(t)=0 \] turns out to be properly stated as well. In the present paper, the authors show that for properly stated linear DAEs with index no greater than 2, the inherent ODE of the original system and that of the adjoint DAE are adjoint to each other if and only if certain characteristic subspaces are independent of \(t\). They additionally consider a reduction of the problem on the basic invariant space, yielding a so-called essentially underlying ODE; through an appropriate choice of the bases in the invariant spaces of the DAE and the adjoint system, the EUODEs are shown to be adjoint to each other. Finally, they define a class of formally selfadjoint DAEs, and characterize the boundary conditions which yield a selfadjoint boundary value problem for the EUODE.