Note on local quadratic growth estimates in bang–bang optimal control problems

Abstract

Strong second-order conditions in mathematical programming play an important role not only as optimality tests but also as an intrinsic feature in stability and convergence theory of related numerical methods. Besides of appropriate firstorder regularity conditions, the crucial point consists in local growth estimation for the objective which yields inverse stability information on the solution. In optimal control, similar results are known in case of continuous control functions, and for bang–bang optimal controls when the state system is linear. The paper provides a generalization of the latter result to bang–bang optimal control problems for systems which are affine-linear w.r.t. the control but depend nonlinearly on the state. Local quadratic growth in terms of L₁ norms of the control variation are obtained under appropriate structural and second-order sufficient optimality conditions.

Keywords:

• bang–bang control structure
• strong local optimality
• sufficient optimality conditions
• solution stability

AMS Subject Classifications:

• 49 K 40
• 49 K 15

Acknowledgments

The author would like to thank Walter Alt and Vladimir Veliov for their long-term encouraging interest in the topic and numerous helpful discussions, and the anonymous referee for his valuable remarks.