Lipschitz stability of broken extremals in bang-bang control problems

Abstract

Optimal bang-bang controls appear in problems where the system dynamics linearly depends on the control input. The principal control structure as well as switching points localization are essential solution characteristics. Under rather strong optimality and regularity conditions, for so-called simple switches of (only) one control component, the switching points had been shown being differentiable w.r.t. problem parameters. In case that multiple (or: simultaneous) switches occur, the differentiability is lost but Lipschitz continuous behavior can be observed e.g. for double switches. The proof of local structural stability is based on parametrizations of broken extremals via certain backward shooting approach. In a second step, the Lipschitz property is derived by means of nonsmooth Implicit Function Theorems. © 2008 Springer.