Exploiting Invariants in the Numerical Solution of Multipoint Boundary Value Problems for DAE

This paper presents a new approach to the numerical solution of boundary value problems for higher index diierential algebraic equations. Invariants known for the original DAE as well as invariants of the reduced index 1 formulation are exploited to stabilize initial value problem integration, derivative generation and nonlinear and linear systems solution of an enhanced multiple shooting method. Extensions to collocation are given. Applications are presented for two important problem classes: parameter estimation in multibody systems given in descriptor form, and singular and state constrained optimal control problems. In particular, generalizations of the \internal numerical diierentiation" technique to DAE with invariants and a new multistage least squares decomposition technique for DAE boundary value problems are developed, which are implemented in the multiple shooting code PARFIT and in the collocation code COLFIT. Further, a method is described which minimizes the number of necessary directional derivatives in the presence of multipoint conditions and invariants. As numerical applications a parameter identiication problem for a slider crank mechanism and a periodic cruise optimal control problem for motor glider aircraft are treated. AMS subject classiications. 65L10, 65L60 0. Introduction. Initial value problems (IVP) for diierential algebraic equations (DAE) have received signiicantly more attention in the previous years than boundary value problems (BVP). In particular, there has been a very rapid development of new integration techniques and software for multibody systems. The present paper concentrates on two important classes of boundary value problems for DAE: parameter estimation in descriptor form models for multibody systems treatment of singular controls and state constraints in optimal control Both problems have in common that they lead to DAE with invariants that arise from index reduction. Additional physical invariants may appear, such as the total energy in conservative mechanical systems or the Hamiltonian in optimal control problems. The focus of this paper is on the numerical exploitation of these invariants in solution algorithms for multipoint boundary value problems. Two diicult application problems are treated in order to demonstrate the resulting beneets. In a parameter estimation problem for a descriptor form multibody system, the condition number of the linear system is reduced, and the number of Gauu-Newton iterations is decreased signiicantly. A family of optimal control problems is solved far beyond the previously reached point along a homotopy path. The numerical solution of nonlinear DAE boundary value problems by multiple shooting exhibits two major additional diiculties as compared to ODE boundary value …