Abstract
L 1- local optimality of a given control ũ(·) in an optimal control problem for an affine control system with bounded controls is investigated. Starting from the Pontryagin Maximum Principle, which is a first-order necessary optimality condition, we develop it in two directions: (1) extending the notions of 1st and 2nd variations of the system along ũ(·), we obtain 1st and 2nd-order sufficient optimality conditions for bang-bang Pontryagin extremals; (2) developing Legendre-Jacobi-Morse-type results for the extended second variation we obtain 2nd-order sufficient optimality conditions for general (bang-bang-singular) type of Pontryagin extremals.
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