Approximation of Linear Control Problems with Bang-Bang Solutions

W. Alt, R. Baier, M. Gerdts, F. Lempio: Approximation of Linear Control Problems with Bang-Bang Solutions
Optimization 62 (1), 9 - 32, 2013, online available in iFirst since May 2011

DOI: 10.1080/02331934.2011.568619
MR Nummer: 3023028
Zentralblattnummer: 1263.49028
Keywords: linear optimal control; bang-bang control; discretization
Mathematics Subject Classification Code: 49J15 (49M25 49N05 49J30)


Abstract:

We analyze the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L1-norm and with order 1/2 w.r.t. the L2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability conditions for these zeros. We also discuss higher-order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

Contents:

1. Introduction
2. Euler Approximation
2.1 Discretization
2.2 Error estimates for the switching function
2.3 Error estimates for bang-bang controls
2.4 Numerical examples
3. Higher Order Approximations

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