Regularization and bang-bang conjugate times in optimal control

Abstract : In this thesis we consider a minimal time control problem for single-input control-affine systems in finite dimension with fixed initial and final conditions, where the scalar control take values on a closed interva1. When applying a shooting method for solving this problem, one may encounter numerical obstacles due to the fact that the shooting function is non smooth whenever the control is bang-bang. For these systems a theoretical concept of conjugate time has been defined in the bang-bang case, however direct algorithms of computation are difficult to apply. Besides, theoretical and practical issues for conjugate time theory are well known in the smooth case, and efficient implementation tools are available. We propose a regularization procedure for which the solutions of the minimal time problem depend on a small enough real positive parameter and are defined by smooth functions with respect to the time variable, facilitating the application of a single shooting method. Under appropriate assumptions, we prove a strong convergence result of the solutions of the regularized problem towards the solution of the initial problem, when the real parameter tends to zero. The conjugate times computation of the locally optimal trajectories for the regularized problem falls into the standard theory. We prove, under appropriate assumptions, the convergence of the first conjugate time of the regularized problem towards the first conjugate time of the initial bang-bang control problem, when the real parameter tends to zero. As a byproduct, we obtain an efficient algorithmic way to compute conjugate times in the bang-bang case.
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Mathematics. Université d'Orléans; Universidade de Aveiro (Portugal), 2010. English. <NNT : 2010ORLE2027>
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Submitted on : Tuesday, April 12, 2011 - 4:56:20 PM
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Cristiana Silva. Regularization and bang-bang conjugate times in optimal control. Mathematics. Université d'Orléans; Universidade de Aveiro (Portugal), 2010. English. <NNT : 2010ORLE2027>. <tel-00585376>

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