H\"older regularity in bang-bang type affine optimal control problems

TL;DR

This paper investigates the H"older regularity of optimality systems in affine control problems, introducing a new control space metric that accounts for bang-bang controls and extending conditions for strong metric sub-regularity to more complex, non-linear problems.

Contribution

It introduces a novel control space metric tailored for bang-bang controls and extends H"older regularity conditions to non-linear state problems with sub-Lipschitz regularity.

Findings

New control space metric captures bang-bang structure

Extended H"older regularity conditions to non-linear problems

Applicable to Model Predictive Control algorithms

Abstract

This paper revisits the issue of H\"older Strong Metric sub-Regularity (HSMs-R) of the optimality system associated with ODE optimal control problems that are affine with respect to the control. The main contributions are as follows. First, the metric in the control space, introduced in this paper, differs from the ones used so far in the literature in that it allows to take into consideration the bang-bang structure of the optimal control functions. This is especially important in the analysis of Model Predictive Control algorithms. Second, the obtained sufficient conditions for HSMs-R extend the known ones in a way which makes them applicable to some problems which are non-linear in the state variable and the H\"older exponent is smaller than one (that is, the regularity is not Lipschitz).