Stability and genericity of bang-bang controls in affine problems
Alberto Domínguez Corella, Gerd Wachsmuth
TL;DR
The paper analyzes stability and genericity of bang-bang controls in affine optimal control problems within a unifying abstract framework that accommodates ODE- and PDE-constrained instances. It establishes a fundamental equivalence between bang-bang structure and strong forms of stability and well-posedness, and shows that bang-bang minimizers are necessary for robust stability properties via growth conditions and subregularity arguments. A central result is that bang-bang strict global minimizers are generic under linear perturbations, derived from Stegall's variational principle and the Radon--Nikodým property. The authors illustrate the theory on affine ODEs, elliptic PDEs, and a velocity-tracking problem, and provide an appendix with revised proofs and extremal-structure results (Visin) to strengthen the mathematical foundations. Collectively, the work clarifies when singular arcs are problematic and when bang-bang regularity ensures well-posedness and stable first-order conditions, with implications for numerical methods and regularization strategies.
Abstract
We analyse the role of the bang-bang property in affine optimal control problems. We show that many essential stability properties of affine problems are only satisfied when minimizers are bang-bang. Moreover, we prove that almost any perturbation in an affine optimal control problem leads to a bang-bang strict global minimizer. We work in an abstract framework that allows to cover many problems in the literature of optimal control, this includes problems constrained by partial and ordinary differential equations. We give examples that show the applicability of our results to specific optimal control problems.