Journal of Convex Analysis 33 (2026), No. 3&4, 801--816
Copyright Heldermann Verlag 2026
About the Bang-Bang Principle for Controlled Affine Dynamics with Brownian Noise
Ruben Chenevat
MISTEA, Université de Montpellier, INRAE, France
ruben.chenevat@inrae.fr
Dan Goreac
(1) Ecole d'Actuariat, Université Laval Québec, Canada
(2) LAMA, Université Gustave Eiffel, UPEM, Univ. Paris Est Creteil, CNRS, Marne-la-Vallée, France
(3) School of Mathematics and Statistics, Shandong University, Weihai, P. R. China
dan.goreac@act.ulaval.ca
Qinlong Li
School of Mathematics and Statistics, Shandong University, Weihai, P. R. China
qinlongli@mail.sdu.edu.cn
Alain Rapaport
MISTEA, Université de Montpellier, INRAE, France
alain.rapaport@inrae.fr
We revisit the Bang-Bang principle, established for deterministic dynamics with affine control, in a stochastic setting where the dynamics are subject to Brownian motion. We show that such a principle does not generally apply in this context. However, we demonstrate that it does apply under certain conditions with deterministic controls. With stochastic controls, we obtain, under certain conditions, that Mayer's problems with an expectation criterion and a compact control set are equivalent to the same problems with controls taking values in the closed convex hull of the control set. This result is illustrated by a linearized dynamics of the SIR epidemiological model.
Keywords: Bang-Bang principle, optimal control, stochastic differential equations, convexity.
MSC: 93E20, 60H10, 49J30.
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