Approximation of generalized minimizers and regularization of optimal control problems

Abstract

An open problem, set by Yu. Orlov in his contribution to the volume "Open Problems in Mathematical Systems and Control Theory", V. Blondel, A. Megretski Eds., 2004, regards regularization of optimal control-affine problems with control-independent state-quadratic cost. It is asked whether the infima of the regularized (by adding squared L 2-norm of controls) functionals converge to the infimum of the original functional? We show that this question can be resolved by an elementary argument. We claim that one should study minimizing sequences of the original functional, rather than its infimum. We advocate the relevance of this question, formulating it via notion of order of singularity of an optimal problem. We study this question and provide computations of the order of singularity for arbitrary singular linear-quadratic problem and also for some classes of nonlinear control-affine problems. Some open problems are set. © Springer-Verlag Berlin Heidelberg 2007.