| This dissertation presents several results from an investigation into the stability and numerical behavior of differential-algebraic equations (DAEs), specifically Lagrangian DAEs. Lagrangian DAEs arise naturally when applying Lagrange's dynamical formulation to a system described with an interdependent coordinate set. The constraint equations that relate the coordinates to each other must be considered in conjunction with the differential relations. A stability theorem is presented for conservative LDAEs by adapting a well-known stability result to the DAE case. Using singular system theory a formula is developed for the characteristic polynomial of a general LDAE in terms of the familiar inertia, damping, stiffness, and constraint Jacobian matrices. With the characteristic polynomial in hand, local stability analysis can proceed as for ordinary differential equations (e.g. the Routh-Hurwitz criteria). A pervasive and difficult problem in the numerical solution of DAEs is the determination of consistent initial conditions. A method is demonstrated, based on perturbation techniques and the analytical solution to linear singular systems, for finding appropriate initial conditions. |
