  This work is concerned with the time optimal control problem for evolution
equations in Hilbert spaces. The attention is focused on the maximum principle
for the time optimal controllers having the dimension smaller that of the state
system, in particular for minimal time sliding mode controllers, which is one
of the novelties of this paper. We provide the characterization of the
controllers by the optimality conditions determined for some general cases. The
proofs rely on a set of hypotheses meant to cover a large class of
applications. Examples of control problems governed by parabolic equations with
potential and drift terms, porous media equation or reaction-diffusion systems
with linear and nonlinear perturbations, describing real world processes, are
presented at the end.
