Appl Math Optim 41:237–254 (2000)
2000 Springer-Verlag New York Inc.
Lipschitzian Regularity of Minimizers for Optimal Control Problems
with Control-Afﬁne Dynamics
A. V. Sarychev and D. F. M. Torres
Department of Mathematics, University of Aveiro,
3810 Aveiro, Portugal
Communicated by J. Stoer
Abstract. We study the Lagrange Problem of Optimal Control with a functional
dt and control-afﬁne dynamics ˙x = f
(a priori) unconstrained control u ∈ R
. We obtain conditions under which the
minimizing controls of the problem are bounded—a fact which is crucial for the
applicability of many necessary optimality conditions, like, for example, the Pon-
tryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian
regularity of minimizers of the Basic Problem of the Calculus of Variations and of
the Problem of the Calculus of Variations with higher-order derivatives.
Key Words. Optimal control, Calculus of variations, Pontryagin Maximum Prin-
ciple, Boundedness of minimizers, Nonlinear control-afﬁne systems, Lipschitzian
AMS Classiﬁcation. 49J15, 49J30.
Under standard hypotheses of the Tonelli existence theory in the Calculus of Variations,
the existence of minimizers is guaranteed in the class of absolutely continuous func-
tions possibly with unbounded derivative. As is known, in such cases the optimality
This research was partially presented at the International Conference dedicated to the 90th Anniversary
of L. S. Pontryagin, Moscow, September 1998.