Appl Math Optim (2013) 68:219–253
Feasible Perturbations of Control Systems with Pure
State Constraints and Applications to Second-Order
Published online: 31 May 2013
© Springer Science+Business Media New York 2013
Abstract We propose second-order necessary optimality conditions for optimal con-
trol problems with very general state and control constraints which hold true under
weak regularity assumptions on the data. In particular the pure state constraints are
general closed sets, the optimal control is supposed to be merely measurable and
the dynamics may be discontinuous in the time variable as well. These results are
obtained by an approach based on local perturbations of the reference process by
second-order tangent directions. This method allows direct and quite simple proofs.
Keywords Optimal control · Pontryagin’s maximum principle · Second-order
necessary optimality conditions · Second-order tangents · Feasible perturbations
Consider the following state constrained control system:
˙x(t) = f (t, x(t), u(t)), x(0) = x
x(t) ∈ K ∀t ∈[0, 1] and u(t) ∈ U(t) for a.e. t ∈[0, 1].
We will assume standard hypotheses on the dynamics f :[0, 1]×R
In the above K is a closed subset of R
and U :[0, 1] R
is a measurable set-
valued map with closed nonempty images. Controls are measurable maps such that
Communicating Editor: Alain Bensoussan.
Financial support by the European Commission (FP7-PEOPLE-2010-ITN, Grant Agreement no.
264735-SADCO) is gratefully acknowledged.
D. Hoehener (
Combinatoire & Optimisation, Institut de Mathématiques de Jussieu (UMR 7586), Université Pierre
et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France