Strong bi-metric regularity in affine optimal control problems
Alberto Domínguez Corella, Marc Quincampoix, Vladimir Veliov
TL;DR
This work addresses the regularity of the optimality map arising from the Pontryagin maximum principle for affine optimal control problems, where both dynamics and cost are linear in the control. It introduces strong bi-metric regularity (Sbi-MR) and provides new sufficient conditions that do not require convexity of the objective, relying on a combined second- and first-order Hamiltonian structure and a symmetry condition $\hat{H}_{ux}(t)\hat{B}(t)$; these yield the Sbi-MR of the optimality map at a reference solution. The proof blends a linearization-based reformulation with a variational inequality framework, showing that the regularity of a linearized map $L$ implies the regularity of $F$, aided by a quadratic-growth condition (via $\Gamma$) and a perturbation analysis around $\hat{\sigma}$. An important consequence is the uniform first-order convergence of Euler discretization for families of affine problems near a reference problem, which has practical relevance for Model Predictive Control. The results also cover special cases, including non-convex objectives and polyhedral control sets, and provide conditions under which the regularity guarantees extend to discretized schemes.
Abstract
The paper presents new sufficient conditions for the property of strong bi-metric regularity of the optimality map associated with an optimal control problem which is affine with respect to the control variable ({\em affine problem}). The optimality map represents the system of first order optimality conditions (Pontryagin maximum principle), and its regularity is of key importance for the qualitative and numerical analysis of optimal control problems. The case of affine problems is especially challenging due to the typical discontinuity of the optimal control functions. A remarkable feature of the obtained sufficient conditions is that they do not require convexity of the objective functional. As an application, the result is used for proving uniform convergence of the Euler discretization method for a family of affine optimal control problems.