  Pontryagin's Maximum Principle is an outstanding result for solving optimal
control problems by means of optimizing a specific function on some particular
variables, the so called controls. However, this is not always enough for
solving all these problems. A high order maximum principle (Krener, 1977) must
be used in order to obtain more necessary conditions for optimality. These new
conditions determine candidates to be optimal controls for a wider range of
optimal control problems.
  Here, we focus on control-affine systems. Krener's high order perturbations
are redefined following the notions introduced in Aguilar-Lewis (2008). A
weaker version of Krener's high order maximum principle is stated in the
framework of presymplectic geometry. As a result, the presymplectic constraint
algorithm in the sense of Gotay-Nester-Hinds (1979) can be used. We establish
the connections between the presymplectic constraint algorithm and the
candidates to be optimal curves obtained from the necessary conditions in
Krener's high order maximum principle. In this paper we obtain weaker geometric
necessary conditions for optimality of abnormal solutions than the ones in
Krener (1977) and the ones in the weak high order maximum principle. These new
necessary conditions are more useful, computationally speaking, for finding
curves candidate to be optimal. The theory is supported by describing
specifically some of the above-mentioned conditions for some mechanical control
systems.
