  In this paper we derive necessary optimality conditions for optimal control
problems with nonlinear and nonsmooth implicit control systems. Implicit
control systems have wide applications including differential algebraic
equations (DAEs). The challenge in the study of implicit control system lies in
that the system may be truly implicit, i.e., the Jacobian matrix of the
constraint mapping may be singular. Our necessary optimality conditions hold
under the so-called weak basic constraint qualification plus the calmness of a
perturbed constraint mapping. Such constraint qualifications allow for
singularity of the Jacobian and hence is suitable for implicit systems.
Specifying these results to control of semi-explicit DAEs we obtain necessary
optimality conditions for control of semi-explicit DAEs with index higher than
one.
