Appl Math Optim 55:255–271 (2007)
2007 Springer Science+Business Media, Inc.
Sufﬁcient Optimality Conditions in Stability Analysis
for State-Constrained Optimal Control
Systems Research Institute, Polish Academy of Sciences,
ul.Newelska 6, 01-447 Warsaw, Poland
Abstract. A family of parametric linear-quadratic optimal control problems is
considered. The problems are subject to state constraints. It is shown that if weak
second-order sufﬁcient optimality conditions and standard constraint qualiﬁcations
are satisﬁed at the reference point, then, for small perturbations of the parameter,
there exists a locally unique stationary point, corresponding to a solution. This point
is a Lipschitz continuous function of the parameter.
Key Words. Linear-quadratic optimal control, State constraints, Parametric prob-
lems, Lipschitzian stability, Second-order sufﬁcient conditions.
AMS Classiﬁcation. 49K40, 49K30, 49K15, 49N10.
Local Lipschitz continuity of solutions to optimal control problems depending on a
parameter has been investigated for about 20 years and for control-constrained problems
the results are fairly complete. In  necessary and sufﬁcient conditions were derived,
under which the solutions to control-constrained optimal control problems have Lipschitz
continuous localizations as functions of parameters. This characterization includes a
constraint qualiﬁcation and a coercivity condition, which must be satisﬁed at the reference
point. These results can be easily extended to mixed control-state constraints (see ).
However, the presence of pure state constraints creates serious difﬁculties. They are
connected with the fact that the properties of solutions to optimal control problems are
This research was supported by the Polish Ministry of Scientiﬁc Research and Information Technology
Grant 3 T11C 051 28.