next up previous
Next: Direct Collocation Method Up: Numerical Methods Previous: Necessary Conditions of

Multiple Shooting Method

The multiple shooting method has shown to be an effective tool in solving highly nonlinear multi-point boundary value problems. The method is described, e. g., by Bulirsch [4] and Stoer, Bulirsch [24]. Its application to a complicated state constrained optimal control problem is described by Bulirsch, Montrone, and Pesch [5]. Here, we used the code BNDSCO due to Oberle [18].
The main drawbacks when applying the multiple shooting method in the numerical solution of optimal control problems are 1. the derivation of the necessary conditions (e. g., the adjoint differential equations), 2. the estimation of the optimal switching structure, and 3. the estimation of an appropriate initial guess of the unknown state and adjoint variables in order to start the iteration process. The great advantage of the multiple shooting method is the verification of the optimality conditions resulting in a highly accurate solution.
To overcome the first drawback a good knowledge of optimal control theory is required. Proper estimates of the switching structure and of the adjoint variables might then be provided by the use of a homotopy or continuation technique. This can be a very laborious task (cf. [5] for an example) that is especially cumbersome in our problem as none of the adjoint variables is given either at 0 or . In this paper we will demonstrate how the drawbacks 2 and 3 can be overcome when a direct collocation method is used in a pre-computation to estimate the optimal switching structure, state and adjoint variables. In the derivation of the necessary conditions we used the symbolic computation method MAPLE due to Char et al. [6].

Oskar von Stryk
Fri Apr 5 21:57:02 MET DST 1996