Stabilization of numerical solutions of boundary value problems exploiting invariants

Abstract

Solving boundary value problems (BVPs) numerically is an important task when dealing with problems of optimal control. In this paper the numerical solution of BVPs for differential algebraic equations (DAEs) is discussed. The method of choice is multiple shooting. Optimal control problems are higher index DAEs in the case of singular controls or state constraints. The common procedure of solving higher index DAEs is to reduce the index by differentiating the algebraic equations until index 1 DAEs or ordinary differential equations (ODEs) are obtained which can be treated directly. Unfortunately, the numerical solution of the index reduced problems often suffers from instabilities introducing a drift from the original algebraic conditions. We interprete the higher index constraints as invariants of the ODE and exploit these invariants in order to improve accuracy, stability and efficiency by a new projection technique. Obviously, conservation properties e.g. for energy or momentum can be used in this sense, but the symplectic structure of Pontryagin's Maximum principle allows for deriving further invariants. Solving the shooting equations by Newton's method requires the computation of sensitivity matrices. This is performed by solving the initial value problems for the variational ODEs together with their invariants or by differentiation of the discretization scheme. The techniques are demonstrated on the example of a flight path optimization problem.