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Presymplectic high order maximum principle

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.