How to verify optimal controls computed by direct shooting methods? - A tutorial

For the solution of optimal control problems, direct methods have been established in the process engineering community. If set up correctly they robustly provide more or less accurate approximations of the exact solution. In the usual engineering practice, neither the distance to the exact solution is reflected, nor the compliance with the continuous necessary conditions in form of Pontryagin's Minimum Principle is checked. At the end, some approximate solution is available but its quality is at question. This tutorial addresses the problem of the verification of optimal controls computed by direct shooting methods. We focus on this popular transcription method though the results are also relevant for other solution strategies. We review known results spread in the mathematical literature on optimal control to show how the output of the nonlinear programs (NLPs) resulting from single shooting transcriptions of optimal control problems can be interpreted in the context of Pontryagin's Minimum Principle. In particular, we show how to approximate continuous adjoint variables by means of the dual information provided by the NLP solver. Based on this adjoint approximation we use a multi-level setting to construct an estimate of the distance to a true extremal solution satisfying the continuous necessary conditions of optimality. A comprehensive case study illustrates the theoretical results. (C) 2011 Elsevier Ltd. All rights reserved.