Differential algebraic equations with properly stated leading term are equations of the form A(x(t),t)(d(x(t),t))'+b(x(t),t)=0 with in some sense well-matched coefficients. Systems resulting from the modified nodal analysis (MNA) in circuit simulation promptly fit into this form. Recent results concerning solvability and numerical treatment of those equations are discussed. An index notion that works via linearization is given. This allows for index criteria just in terms of the coefficients A,d,b and their first partial derivatives, no further derivative arrays are used.
The index of DAE systems arising from linear quadratic optimal control problems is considered. Necessary and sufficient conditions ensuring regularity with tractability index one are proved. Then, it is shown that if the control problem DAE is regular with index one and if the leading term of the DAE to be controlled is given by one full-column-rank and one full-row-rank matrix, then it has a Hamiltonian inherent explicit ODE. For problems with regular index zero or index one DAEs to be controlled, the DAE of the control problem is shown to be regular with tractability index one or three, depending on whether the control coefficient R is singular.
By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint equation. When applied to boundary value problems, the result provides a constructive criterion for unique solvability and allows reducing the problem to initial value problems and linear algebraic equations.
Models for physical systems often take the form of implicit or behavioral models. One important problem is the
identification of which combinations of variables are good candidtates for control variables. This paper first provides one
solution to this problem for linear time varying systems. The solution is shown to be related to a general optimization
problem. It is then shown how these same algorithms can be extended to a large and important class of nonlinear systems.