Appl Math Optim 40:79–103 (1999)
1999 Springer-Verlag New York Inc.
Maximum Principle for Implicit Control Systems
E. N. Devdariani
and Yu. S. Ledyaev
Department of Mathematics and Statistics, Queen’s University,
Kingston, Ontario, Canada K7L 3N6
Steklov Institute of Mathematics,
Moscow 117966, Russia
Communicated by J. Stoer
Abstract. In this paper we derive optimality conditions in the form of a maxi-
mum principle for an optimal control problem for nonlinear implicit, or descriptor,
control systems. The regularity conditions which are imposed on the system allow
us to reduce the optimal control problem to an equivalent nonsmooth variational
one. Nonsmooth analysis techniques together with the sliding variations of relaxed
controls from optimal control are used to obtain necessary conditions.
Key Words. Optimal control, Differential inclusion, Nonsmooth analysis, Maxi-
AMS Classiﬁcation. 49K15.
In this paper we consider the optimal control problem of minimizing a functional over
trajectories x(t) and corresponding controls u(t) of an implicit control system.
An implicit control system is a dynamical system where the current velocity ˙x(t ) of
the state vector is deﬁned implicitly by the current state vector x(t) and the control u(t).
This work was partially supported by the Russian Fund of Fundamental Research under Grant 93-011-
16032 and by the Natural Sciences Engineering Research Council of Canada and Fonds pour la Formation de
Chercheurs et l’Aide `a la Recherche du Qu´ebec. The current address of the second author is the Department
of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008, USA.