Hölder regularity in bang-bang type affine optimal control problems
Alberto Domínguez Corella, Vladimir Veliov
TL;DR
The paper studies Hölder strong metric sub-regularity (HSMs-R) of the optimality system for affine-in-control ODE optimal control problems, introducing a new metric on the control space that reflects bang-bang structure and is particularly relevant for Model Predictive Control. It formalizes the problem in terms of the spaces $\mathcal{Y}$ and $\mathcal{Z}$ and the optimality map $\Phi:\mathcal{Y}\to\mathcal{Z}$, and proves that under standard Assumptions 1–3, $\Phi$ is Hölder SMs-R with exponent $1/\nu^2$ at the reference $\hat y$. The main result provides explicit a priori constants $a,b,\kappa$ ensuring that small residual perturbations imply a Hölder-type proximity between feasible and reference solutions with rate $d_{\mathcal{Y}}(y,\hat y) \le \kappa d_{\mathcal{Z}}(z,\hat z)^{1/\nu^2}$. The proofs hinge on two lemmas giving switching-function bounds and a Grönwall-based stability estimate, and an illustrative example demonstrates the framework in a bang-bang, nonlinear-in-state setting and clarifies the sufficiency of Assumptions 2–3 for such problems.
Abstract
This paper revisits the issue of Hölder 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 Hölder exponent is smaller than one (that is, the regularity is not Lipschitz).